Topic 10 Further Applications of Integration

10.1 Arc Lengths and Areas of Surfaces of Revolutions

• Arc length:

  The length $L$ of an arc: $y=f\left(x\right)$, $a\le x\le b$ is $$L={\int }_a^b\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx={\int }_{f\left(a\right)}^{f\left(b\right)}\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dy.$$

• Surface area of a revolution

  The area $S$ of the surface rotating an arc: $y=f\left(x\right)$, $a\le x\le b$ about the $x$-axis is $$S=2\pi \int rds=2\pi \int yds,$$ and about the $y$-axis is $$S=2\pi \int rds=2\pi \int xds,$$ where $$ds=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dx=\sqrt{1+{\left(\frac{dy}{dx}\right)}^2}dy.$$ The integral limits depend on whether you use $dx$ or $dy$ in the integral.

In Maple, the package Student[Calculus1] provides commands to investigate arc length and surface area of revolutions: ArcLength(f(x), x = a..b, opts)} SurfaceOfRevolution(f(x), x = a..b, opts)}

Example 10.1 Set up an integral and evaluate the integral for the length of the curve defined by $$f\left(x\right)=\sqrt{x},1\le x\le 4.$$ Plot $f\left(x\right)$ together with the arc length function in the same coordinate system.

Solution.

# Load the package

with(Student[Calculus1])

# Set up an integral

ArcLength(sqrt(x),x=1..4,output=integral)

# Evaluate the integral

ArcLength(sqrt(x),x=1..4)

# Plot the function and the arc length function

ArcLength(sqrt(x), x=1..4, output=plot)

Example 10.2 Set up an integral and evaluate the integral for the area of the surface obtained by rotating the curve defined by $$f\left(x\right)=\sqrt{x},1\le x\le 4$$ about the $y$-axis. Plot the surface of the revolution.

Solution.

# Load the package (skip if the package was already loaded)

with(Student[Calculus1])

# Plot the surface

SurfaceOfRevolution(sqrt(x), x=1..4, output=plot, axis=vertical)

# Set up an integral

SurfaceOfRevolution(sqrt(x),x=1..4,output=integral, axis=vertical)

# Evaluate the integral

SurfaceOfRevolution(sqrt(x),x=1..4, axis=vertical)

Exercise 10.1 Set up an integral and evaluate the integral for the length of the arc defined by $$f\left(x\right)=\ln x,1\le x\le 2.$$ Plot $f\left(x\right)$ together with the arc length function in the same coordinate system.

Exercise 10.2 Plot the surface obtained by rotating the curve defined by $$f\left(x\right)=\frac{\cos x}{x},0\le x\le 4\pi$$ about the $y$-axis. Set up an integral for the area of the surface.

Exercise 10.3 Find the area of the surface obtained by rotating the curve defined by $$f\left(x\right)=\sqrt{1+x^2},0\le x\le 3.$$