Modules › Volumes, arc length and surface area
Volumes, arc length and surface area
Volumes by cross-sections, disks, washers and shells; arc length; area of a surface of revolution.
Lecture 8 · Tutorial 8 · Definite integrals and their applications
Textbook sections: Thomas ch. 5 Integrals (5.1–5.6) and ch. 6 (6.1–6.4) · Stewart ch. 5 (5.1–5.5), ch. 6 (6.1–6.3, 6.5) and ch. 8 (8.1–8.2)
Textbook sections: Thomas ch. 5 Integrals (5.1–5.6) and ch. 6 (6.1–6.4) · Stewart ch. 5 (5.1–5.5), ch. 6 (6.1–6.3, 6.5) and ch. 8 (8.1–8.2)
Volumes, arc length and surface area
Calculus I · Module 9
After this module you can
- Set up volumes by cross-sections, disks/washers and cylindrical shells and choose the easier method.
- Compute arc length and surface area of revolution when the integrand simplifies to a perfect square.
- Cross-check a volume (Pappus' theorem, a quick estimate) and recognise the standard setups of Tutorial 8.
Volumes: three methods
| Method | Formula | Use when |
|---|---|---|
| cross-sections | $V=\int A(x)\,dx$ | known shape of slices |
| disk / washer (about the $x$-axis) | $\pi\int(R^2-r^2)\,dx$ | slices ⊥ axis of rotation |
| shells (about the $y$-axis) | $2\pi\int x\,h(x)\,dx$ | slices ∥ axis of rotation |
Radius of a shell = distance to the axis; for the axis $x=4$ it is $4-x$.
Worked setups
- Disks with diameter from $y=x^2$ to $y=2-x^2$ on $[-1,1]$: $V=\pi\int(1-x^2)^2dx=\frac{16\pi}{15}$.
- Triangle $2y=x+4$, $y=x$, $x=0$: about the $x$-axis (washers) $16\pi$; about the $y$-axis (shells) $\frac{32\pi}3$; about $x=4$ (shells) $\frac{64\pi}{3}$; about $y=8$ (washers) $48\pi$.
- Region between $y=4-x^2$ and $y=2-x$ about the $x$-axis: $\pi\int_{-1}^2[(4-x^2)^2-(2-x)^2]dx=\frac{108\pi}5$.
- Check with Pappus: $V=2\pi\cdot(\text{distance of the centroid})\cdot(\text{area})$.
Arc length and surface area
$$L=\int_a^b\sqrt{1+f'(x)^2}\,dx,\qquad S=2\pi\int_a^bf(x)\sqrt{1+f'(x)^2}\,dx$$
- $y=\frac{x^3}3+\frac1{4x}$ on $[1,3]$: $1+y'^2=\left(x^2+\frac1{4x^2}\right)^2$, $L=\frac{53}6$.
- $y=\int_0^x\tan t\,dt$: $\sqrt{1+\tan^2x}=\sec x$, $L=\ln\sqrt3$ on $[0,\frac\pi6]$.
- Astroid $x^{2/3}+y^{2/3}=1$: total length $6$.
Typical mistakes
Wrong radiusRadius is the distance to the axis, not the coordinate.
Washer vs diskSubtract the squares $R^2-r^2$, not $(R-r)^2$.
Arc lengthLook for the perfect square under the root.
Practice next
- Chapter quizzes Ch. 4 (3 sets)
- Practice bank, filtered to this chapter, then written mode
- Lab: Riemann sums, trapezoid and Simpson
- Then the mock exams (Exams page)
Worked examples from the course problems
Worked example 1 · Volumes by cross-sections (Tutorial 8, Q1)
A solid lies between $x=-1$ and $x=1$; its cross-sections perpendicular to the $x$-axis are disks with diameter from $y=x^2$ to $y=2-x^2$. Its volume is
Show the answer and the solution
Answer: $\frac{16 \pi}{15}$
Diameter $d=2-2x^2$, radius $1-x^2$: $V=\pi\int_{-1}^1(1-x^2)^2dx=\pi\left(2-\frac43+\frac25\right)=\frac{16\pi}{15}$.
Worked example 2 · Volumes of revolution (Tutorial 8, Q6(a))
Revolve the triangle bounded by $2y=x+4$, $y=x$ and $x=0$ about the $x$-axis (washer method). Volume:
Show the answer and the solution
Answer: $16 \pi$
$V=\pi\int_0^4\left[\left(\frac{x+4}{2}\right)^2-x^2\right]dx=\pi\int_0^4\left(4+2x-\frac34x^2\right)dx=16\pi$.
Worked example 3 · Arc length and surface area (Tutorial 8, Q7)
Find the length of $y=\dfrac{x^3}{3}+\dfrac{1}{4x}$ for $1\le x\le3$.
Show the answer and the solution
Answer: $\frac{53}{6}$
$y'=x^2-\frac1{4x^2}$ and $1+y'^2=\left(x^2+\frac1{4x^2}\right)^2$, so $L=\int_1^3\left(x^2+\frac1{4x^2}\right)dx=\left[\frac{x^3}{3}-\frac1{4x}\right]_1^3=\frac{53}6$.
Worked example 4 · Volumes by cross-sections (Tutorial 8, Q2(a))
The base of a solid is the region under $y=2\sqrt{\sin x}$, $0\le x\le\pi$; cross-sections perpendicular to the $x$-axis are equilateral triangles with a side on the base. Volume:
Show the answer and the solution
Answer: $2 \sqrt{3}$
Side $s=2\sqrt{\sin x}$, area $\frac{\sqrt3}4s^2=\sqrt3\sin x$; $V=\sqrt3\int_0^\pi\sin x\,dx=2\sqrt3$.
Practice
Quiz set A Quiz set B Quiz set C Practice bank (this chapter) Written mode Lab: Riemann sums, trapezoid and Simpson