Calculus

Calculus

Function

What is Function?

Introduction to Calculus: What is Derivative and Integral

This is a topic ties everything about (functions and graphs) together.

We're mainly concerned with two parts:

1. Derivative (Differentiation)
2. Integrals (Integration)

A derivative is equal to the rate of change. Most of the time we will use $dy \over dx$ to present how one variable changes with another. The derivative is the gradient of a tangent line.

But from the beginning, we are going to talk about where do you want the derivative, the other way to say is I will give you a point and then tell me what's the derivative or gradient for that point.

From the beginning here are the concepts you need for calculus.

1. "what is the derivative at x=n" you can understand as "what is the gradient when x=n".
2. "What is the integral at x=a to x=b" you can understand as "what is the area between the function and the x-axis from x=a to x=b".

Conclusion (from the beginning )

derivative = gradient of a tangent = rate of change.

Integral = area under the function.

Differentiation

Limits

$\lim _{x\to a}f(x)=L,$

Properties of Limits

1. Sum Rule: The limit of the sum of two functions is the sum of their limits : $\lim _{x\to c}(f(x)+g(x))=\lim _{x\to c}(f(x))+\lim _{x\to c}(g(x))$
2. Difference Rule: The limit of the difference of two functions is the difference of their limits : $\lim _{x\to c}(f(x)-g(x))=\lim _{x\to c}f(x)-\lim _{x\to c}g(x)$
3. Product Rule: The limit of a product of two functions is the product of their limits : $\lim _{x\to c}(f(x)*g(x))=\lim _{x\to c}f(x)*\lim _{x\to c}g(x)$
4. Constant Multiple Rule: The limit of a constant times a function is the constant times the limit of the function :

$\lim _{x\to c}(k*g(x))=k*\lim _{x\to c}g(x)$

1. Quotient Rule: The limit of a quotient of two functions is the quotient of their limits, provided the limit of the denominator is not zero. $\lim _{x\to c}{(f(x) \over g(x))}={\lim _{x\to c}f(x) \over \lim _{x\to c}g(x)}$

Power Rule

$(cx^{n})'=c*n*x^{n-1}$

Derivative of Polynomial Functions

=Newton Derivative of Polynomial Functions=

1. The sum rule $(f+g)'=f'+g'$
2. The Difference Rule $(f-g)'=f'-g'$
3. The Product Rule $(f*g)'=f*g'+g*f'$
4. The Quotient Rule $({f \over g})'={(gf'-fg') \over g^{2}}$

=Leibniz Derivative of Polynomial Functions=

1. The sum rule ${d(f+g) \over dx}={df \over dx}+{dg \over dx}$
2. The Difference Rule ${d(f-g) \over dx}={df \over dx}-{dg \over dx}$
3. The Product Rule ${d(fg) \over dx}'=f{dg \over dx}+g{df \over dx}$
4. The Quotient Rule ${d({f \over g}) \over dx}={g{df \over dx}-f{dg \over dx} \over g^{2}}$

Derivative of Trigonometric Functions

1. $f(sinx)'=cos(x)$
2. $f(cosx)'=-sin(x)$
3. $f(tanx)'=sec^{2}(x)$
4. $f(cotx)'=-csc^{2}(x)$
5. $f(cscx)'=-csc(x)cot(x)$
6. $f(secx)'=sec(x)tan(x)$

Chain Rule

1. Newton Chain rule $[f(g(x))]'=f'(g(x))*g'(x)$
2. Leibniz Chain rule ${\frac {df(g(x))}{dx}}={{df(g)} \over dg}{dg \over dx}$

Derivatives of Logarithmic and Exponential Functions

1. Natural log of $x$ : $(ln(x))'={1 \over x}*(x)'$
2. log of $x$ : $(log_{a}x^{z})'=({lnx^{z} \over lna})'={1 \over ln_{a}}(lnx^{z})'={zx^{z-1} \over ln{a*x^{z}}}$

What is Log

1. First law If $log_{a}x=n$ than $a^{n}=x$
2. Second law $log_{n}A+log_{n}B=log_{(}AB)$
3. Thirde law $log_{A}-log_{B}=log_{(}{A \over B})$
4. law If "${log_{x}}^{log_{x}}={log_{x}}^{2}=n$" than "${log_{x}}={\sqrt[{2}]{n}}$"
5. law if $log_{(}x^{n})={log_{x}}^{n}=n*{log_{x}}$
6. law "$log_{n}(x)={log_{n}(a)}$" than "x = a"

The Exponential Function

1. Exponential $e=1+1+{1 \over 2!}+{1 \over 3!}+{1 \over 4!}+{1 \over 5!}+{1 \over 6!}+{1 \over 7!}....$
2. Exponential Function $e^{x}=1+x+{x^{2} \over 2!}+{x^{3} \over 3!}+{x^{4} \over 4!}+{x^{5} \over 5!}+{x^{6} \over 6!}+{x^{7} \over 7!}...$
3. Exponential Function $(e^{x})'=e^{x}$

so when every you meet Exponential just reminds yourself it will not be changed by the derivatives.

Higher Derivatives

1. Second derivatives $f''(x)=(f'(x))'$

Newton Higher Derivatives

1. function$f(x)$
2. first derivative $f'(x)$
3. Second derivatives $f''(x)=(f'(x))'$
4. Third derivatives $f'''(x)=(f''(x))'$
5. 4th derivatives $f^{(4)}(x)=(f'''(x))'$

Leibniz Higher Derivatives

1. function $y=f(x)$
2. first derivative ${dy} \over {dx}$
3. second derivatives ${d^{2}}y \over {dx^{2}}$
4. third derivatives ${d^{3}}y \over {dx^{3}}$
5. 4th derivatives ${d^{4}}y \over {dx^{4}}$

Why do we need Higher Derivatives

From the zero derivative to sixth derivative there is a meaning on the graph

1. No derivative function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } = Position
2. first derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x) } = Velocity
3. second derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f''(x) } = Acceleration
4. third derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'''(x) } = Jerk
5. fourth derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(4)}(x)} = Snap
6. fifth derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(5)}(x)} = crackle/flounce
7. sixth derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(6)}(x)} = Pop

How to finding Local Maxima and Minima by Differentiation

Our example will be : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)= x^3-12x+1 } And the graph will be the one at the side.

Then we will need to use the derivatives to find the two-point that has 0 slopes we will also call it Local maxima and Local minima

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=3x^2-12 }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(x)=3x^2-12}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3x^2=12}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2=4 }

Then we get x=2 and x=-2 so then we put them into the function.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)= 2^3-(12*2)+1=-15}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)= -2^3-(12*-2)+1= 17}

so the first point will be local maxima it will be at (-2,17) and then local minima (2,-15).

Graphing Functions and Their Derivatives

1. When a first derivative is positive, the original function is increasing.
2. When a first derivative is negative, the original function is decreasing.
   1. The derivative does not tell us if the function is positive or negative, only if increasing or decreasing.
3. When a second derivative is positive, the original function is concave up.
4. When a second derivative is negative, the original function is concave down.
5. When a first derivative hits zero from below the axis, the original function is at a(n) local minimum.
6. When a second derivative is zero, the original function is at a(n) inflection point.

Limits and L'Hospital's Rule

1. L'Hospital's Rule 1 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x\to 0} {f(x) \over g(x)} = lim_{x\to 0} {f'(x) \over g'(x)} }
2. L'Hospital's Rule 2 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \lim_{x\to \infty } {f(x) \over g(x)} = lim_{x\to \infty} {f'(x) \over g'(x)} }

Integration

Mathematicians have been playing with the concept of integration for ages, but it was until Newton’s time that it was realized that integration and differentiation are inverse operations.

Integral

1. Indefinite Integral $\int f(x)\,dx=F(x)+c,$
2. sum rule of Indefinite Integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int [f(x)+g(x)] \,dx = \int f(x) \,dx + \int g(x) \,dx }
3. The Difference Rule Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int [f(x)-g(x)] \,dx = \int f(x) \,dx - \int g(x) \,dx}
4. Indefinite Integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^n \,dx = { x^{n+1} \over n+1 }+c}
5. Natural log rule $\int {n \over x}\,dx={ln|x^{n}|}$
6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int a^x dx = {a^x \over ln (a)}}
7. constant(constant can be pull out in the Indefinite Integral) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int c* f(x)dx = c \int f(x)dx}

Properties of Indefinite Integral

sum rule of Indefinite Integral

1. sum rule of Indefinite Integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int [f(x)+g(x)] \,dx = \int f(x) \,dx + \int g(x) \,dx }

1. The Difference Rule Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int [f(x)-g(x)] \,dx = \int f(x) \,dx - \int g(x) \,dx}

Integrals With Trigonometric Functions

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int cosx \,dx = sinx + c }
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int sin x \,dx = -cos x + c }
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {sec}^2 x \,dx = tan x + c }
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int {csc}^2 x \,dx = -cotx + c }
5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int secx \, tanx \,dx = secx + c }
6. $\int cscx\,cotx\,dx=-cscx+c$

Integration By Parts

1. Performing Integration By Parts $\int f(x)g'(x)\,dx=f(x)g(x)-\int f'(x)g(x)\,dx$

1. Performing Integration By Parts Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int u \,dv = uv - \int v \,du}

All equations that you may need to know for integration

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int x^n \,dx = { x^{n+1} \over n+1 } }
2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int 1 \over x \,dx = { ln |x| } }
3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int a^x \,dx = { a^x \over ln a } }
4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int e^x \,dx = e^x }
5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int sin x \,dx = -cos x }
6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int cos x \,dx = sin x }
7. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int sec x tan x \,dx = sec x }
8. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int sec^2 x \,dx = tan x }
9. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int csc^2 x \,dx = -cot x }
10. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \csc x \cot x \,dx = -csc x }
11. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int sec x \,dx = ln |sec x + tan x|}
12. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int csc x \,dx = ln |csc x - cot x|}
13. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int tan x \,dx = ln |sec x|}
14. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int cot x \,dx = ln |sin x|}

Definite integrals area under a curve and above the curve

1. Definite Integral under and above the curve Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{a}^{b} f(x) \,dx, \int_{b}^{c} |f(x)| \,dx = F(b) - F(a) + |F(b) - F(a)|,}

Area between curves

$\int _{a}^{b}(upperfunction-lowerfunction)\,dx,$

if f(x) = upper function and g(x) = lower function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{a}^{b} (f(x) - g(x)) \,dx,}

Kinematics

Kinematics is a type of physics

Kinematics is a physics topic taking about how describes the motion of points, objects without considering the forces that cause them to move. In Kinematics we will use calculus a lot of times, by using calculus can be a kind of transformation of the graph.

So it will be required Higher Derivatives and Integration.

derivative and integration

1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t)=} displacement (m)
2. first derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s'(t)= a(t)=} velocity (m/s)
3. second derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s''(t)= v(t)=} acceleration
4. third derivative Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s'''= s(t)=} time (second or s )

Same integration

1. no integration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=} time (second or s )
2. first integration of t Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(t)=} acceleration
3. second integration of t Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(t)=} velocity (m/s
4. third integration of t Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s(t)=} displacement (m)

Using Integration to calculate volume

This time we will start to learn how to use integration to calculate the volume.

Now we will start with a function:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)= \sqrt x}

We will start to calculate the volume of the function when the function rotates 360 degrees on the x-axis.

It will never leave the x-axis, and then it will form a 3-dimensional volume. We will call this solid of revolution because we obtain it by revolving a region about a line.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)= \sqrt x} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{0}^{1} A(x)dx}

so the radius of a circle will be r and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \sqrt x}

and we can think of it as the volume is made of many disks and it will form this volume.

So the area of the disk will be like this $A=\pi r^{2}$

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \sqrt x}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r^2 = x}

so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = \pi {(\sqrt x)}^2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(x) = \pi x}

so what we will get this

$\int _{0}^{1}\pi xdx$

so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi } is a constant so we can pull it out

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi \int_{0}^{1} x dx}

Reference

1. IB Maths AA SL/HL calculus
2. Calculus by Professor Dave Explains