# Topic 10 Further Applications of Integration

## 10.1 Arc Lengths and Areas of Surfaces of Revolutions

• Arc length:

The length $$L$$ of an arc: $$y=f(x)$$, $$a\leq x \leq b$$ is $L=\int_a^b\sqrt{1+\left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^2}~\mathrm{d} x=\int_{f(a)}^{f(b)}\sqrt{1+\left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^2}~\mathrm{d} y.$

• Surface area of a revolution

The area $$S$$ of the surface rotating an arc: $$y=f(x)$$, $$a\leq x \leq b$$ about the $$x$$-axis is $S=2\pi\int r\mathrm{d} s = 2\pi\int y\mathrm{d} s,$ and about the $$y$$-axis is $S=2\pi\int r\mathrm{d} s = 2\pi\int x\mathrm{d} s,$ where $\mathrm{d} s=\sqrt{1+\left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^2}~\mathrm{d} x = \sqrt{1+\left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)^2}~\mathrm{d} y.$ The integral limits depend on whether you use $$\mathrm{d} x$$ or $$\mathrm{d}y$$ 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 $f(x)=\sqrt{x},\qquad 1\leq x\leq 4.$ Plot $$f(x)$$ 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 $f(x)=\sqrt{x},\qquad 1\leq x\leq 4$ 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 $f(x)=\ln x, \qquad 1\leq x\leq 2.$ Plot $$f(x)$$ together with the arc length function in the same coordinate system.

Exercise 10.2 Plot the surface obtained by rotating the curve defined by $f(x)=\frac{\cos x}{x}, \qquad 0\leq x\leq 4\pi$ 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 $f(x)=\sqrt{1+x^2},\qquad 0\leq x\leq 3.$