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 \[ f(x)=\sqrt{1+x^2},\qquad 0\leq x\leq 3. \]