• Home
  • Textbooks
  • Mathematical Methods for Physics and Engineering: A Comprehensive Guide
  • Multiple integrals

Mathematical Methods for Physics and Engineering: A Comprehensive Guide

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 6

Multiple integrals - all with Video Answers

Educators

Chapter Questions

05:32

Problem 1

Sketch the curved wedge bounded by the surfaces $y^{2}=4 a x, x+z=a$ and $z=0$, and hence calculate its volume $V$.

Mehdi Hatefipour
Mehdi Hatefipour
Numerade Educator
03:55

Problem 2

Evaluate the volume integral of $x^{2}+y^{2}+z^{2}$ over the rectangular parallelepiped bounded by the six surfaces $x=\pm a, y=\pm b, z=\pm c$.

Jacob Fry
Jacob Fry
Numerade Educator
06:24

Problem 3

Find the volume integral of $x^{2} y$ over the tetrahedral volume bounded by the planes $x=0, y=0, z=0$, and $x+y+z=1$

Mehdi Hatefipour
Mehdi Hatefipour
Numerade Educator
View

Problem 4

Evaluate the surface integral of $f(x, y)$ over the rectangle $0 \leq x \leq a, 0 \leq y \leq b$ for the functions
(a) $f(x, y)=\frac{x}{x^{2}+y^{2}}$,
(b) $f(x, y)=(b-y+x)^{-3 / 2}$.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 5

(a) Prove that the area of the ellipse
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
$$
is $\pi a b$.
(b) Use this result to obtain an expression for the volume of a slice of thickness $d z$ of the ellipsoid
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1
$$
Hence show that the volume of the ellipsoid is $4 \pi a b c / 3$.
210

Eduard Sanchez
Eduard Sanchez
Numerade Educator
05:19

Problem 6

The function
$$
\Psi(r)=A\left(2-\frac{Z r}{a}\right) e^{-Z r / 2 a}
$$
gives the form of the quantum mechanical wavefunction representing the electron in a hydrogen-like atom of atomic number $Z$ when the electron is in its first allowed spherically symmetric excited state. Here $r$ is the usual spherical polar coordinate, but, because of the spherical symmetry, the coordinates $\theta$ and $\phi$ do not appear explicitly in $\Psi$. Determine the value that $A$ (assumed real) must have if the wavefunction is to be correctly normalised, i.e. the volume integral of $|\Psi|^{2}$ over all space is equal to unity.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:31

Problem 8

A planar figure is formed from uniform wire and consists of two semicircular arcs, each with its own closing diameter, joined so as to form a letter ' $B$ '. The figure is freely suspended from its top left-hand corner. Show that the straight edge of the figure makes an angle $\theta$ with the vertical given by $\tan \theta=(2+\pi)^{-1}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
View

Problem 9

A certain torus has a circular vertical cross-section of radius $a$ centred on a horizontal circle of radius $c(>a)$.
(a) Find the volume $V$ and surface area $A$ of the torus, and show that they can be written as
$$
V=\frac{\pi^{2}}{4}\left(r_{o}^{2}-r_{\mathrm{i}}^{2}\right)\left(r_{\mathrm{o}}-r_{\mathrm{i}}\right), \quad A=\pi^{2}\left(r_{o}^{2}-r_{\mathrm{i}}^{2}\right)
$$
where $r_{0}$ and $r_{o}$ are respectively the outer and inner radii of the torus.
(b) Show that a vertical circular cylinder of radius $c$, coaxial with the torus, divides $A$ in the ratio
$$
\pi c+2 a: \pi c-2 a
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
07:07

Problem 10

A thin uniform circular disc has mass $M$ and radius $a$.
(a) Prove that its moment of inertia about an axis perpendicular to its plane and passing through its centre is $\frac{1}{2} M a^{2}$.
(b) Prove that the moment of inertia of the same disc about a diameter is $\frac{1}{4} \mathrm{Ma}^{2}$.
This is an example of the general result for planar bodies that the moment of inertia of the body about an axis perpendicular to the plane is equal to the sum
211of the moments of inertia about two perpendicular axes lying in the plane: in an obvious notation
$$
I_{z}=\int r^{2} d m=\int\left(x^{2}+y^{2}\right) d m=\int x^{2} d m+\int y^{2} d m=I_{y}+I_{x}
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
06:04

Problem 11

In some applications in mechanics the moment of inertia of a body about a single point (as opposed to about an axis) is needed. The moment of inertia $I$ about the origin of a uniform solid body of density $\rho$ is given by the volume. integral
$$
I=\int_{V}\left(x^{2}+y^{2}+z^{2}\right) \rho d V
$$
Show that the moment of inertia of a right circular cylinder of radius $a$, length $2 b$, and mass $M$ about its centre is
$$
M\left(\frac{a^{2}}{2}+\frac{b^{2}}{3}\right)
$$

Mehdi Hatefipour
Mehdi Hatefipour
Numerade Educator
View

Problem 12

The shape of an axially symmetric hard-boiled egg, of uniform density $\rho_{0}$, is given in spherical polar coordinates by $r=a(2-\cos \theta)$, where $\theta$ is measured from the axis of symmetry.
(a) Prove that the mass $M$ of the egg is $M=\frac{40}{3} \pi \rho_{0} a^{3}$.
(b) Prove that the egg's moment of inertia about its axis of symmetry is $\frac{342}{175} \mathrm{Ma}^{2}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
View

Problem 13

In spherical polar coordinates $r, \theta, \phi$ the element of volume for a body that is symmetrical about the polar axis is $d V=2 \pi r^{2} \sin \theta d r d \theta$, whilst its element of surface area is $2 \pi r \sin \theta\left[(d r)^{2}+r^{2}(d \theta)^{2}\right]^{1 / 2} .$ A particular surface is defined by $r=2 a \cos \theta$, where $a$ is a constant, and $0 \leq \theta \leq \pi / 2 .$ Find its total surface area and the volume it encloses, and hence identify the surface.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
04:20

Problem 14

By expressing both the integrand and the surface element in spherical polar coordinates, show that the surface integral
$$
\int \frac{x^{2}}{x^{2}+y^{2}} d S
$$
over the surface $x^{2}+y^{2}=z^{2}, 0 \leq z \leq 1$, has the value $\pi / \sqrt{2}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
View

Problem 15

By transforming to cylindrical polar coordinates, evaluate the integral
$$
I=\iiint \ln \left(x^{2}+y^{2}\right) d x d y d z
$$
over the interior of the conical region $x^{2}+y^{2} \leq z^{2}, 0 \leq z \leq 1$

Victor Salazar
Victor Salazar
Numerade Educator
08:20

Problem 16

Sketch the two families of curves
$$
y^{2}=4 u(u-x), \quad y^{2}=4 v(v+x)
$$
where $u$ and $v$ are parameters.
By transforming to the uv-plane evaluate the integral of $y /\left(x^{2}+y^{2}\right)^{1 / 2}$ over that part of the quadrant $x>0, y>0$ bounded by the lines $x=0, y=0$ and the curve $y^{2}=4 a(a-x)$.

Leon Druch
Leon Druch
Numerade Educator
View

Problem 17

By making two successive simple changes of variables, evaluate
$$
I=\iiint x^{2} d x d y d z
$$
over the ellipsoidal region
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} \leq 1
$$

Victor Salazar
Victor Salazar
Numerade Educator
02:25

Problem 18

Sketch the domain of integration for the integral
$$
I=\int_{0}^{1} \int_{x-y}^{1 / y} \frac{y^{3}}{x} \exp \left[y^{2}\left(x^{2}+x^{-2}\right)\right] d x d y
$$
and characterise its boundaries in terms of new variables $u=x y$ and $v=y / x$. Show that the Jacobian for the change from $(x, y)$ to $(u, v)$ is equal to $(2 v)^{-1}$, and hence evaluate $I$.

Sam Stansfield
Sam Stansfield
Numerade Educator
02:25

Problem 19

Sketch the domain of integration for the integral
$$
I=\int_{0}^{1} \int_{x-y}^{1 / y} \frac{y^{3}}{x} \exp \left[y^{2}\left(x^{2}+x^{-2}\right)\right] d x d y
$$
and characterise its boundaries in terms of new variables $u=x y$ and $v=y / x$. Show that the Jacobian for the change from $(x, y)$ to $(u, v)$ is equal to $(2 v)^{-1}$, and hence evaluate $I$.

Sam Stansfield
Sam Stansfield
Numerade Educator
View

Problem 20

Define a coordinate system $u, v$ whose origin coincides with that of the usual $x, y$ system and whose $u$-axis coincides with the $x$-axis, whilst the $v$-axis makes an angle $\alpha$ with it. By considering the integral $I=\int \exp \left(-r^{2}\right) d A$, where $r$ is the radial distance from the origin, over the area defined by $0 \leq u<\infty, 0 \leq v<\infty$, prove that.
$$
\int_{0}^{\infty} \int_{0}^{\infty} \exp \left(-u^{2}-v^{2}-2 u v \cos \alpha\right) d u d v=\frac{\alpha}{2 \sin \alpha}
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
View

Problem 21

Define a coordinate system $u, v$ whose origin coincides with that of the usual $x, y$ system and whose $u$-axis coincides with the $x$-axis, whilst the $v$-axis makes an angle $\alpha$ with it. By considering the integral $I=\int \exp \left(-r^{2}\right) d A$, where $r$ is the radial distance from the origin, over the area defined by $0 \leq u<\infty, 0 \leq v<\infty$, prove that.
$$
\int_{0}^{\infty} \int_{0}^{\infty} \exp \left(-u^{2}-v^{2}-2 u v \cos \alpha\right) d u d v=\frac{\alpha}{2 \sin \alpha}
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
View

Problem 22

As stated in section 5.11, the first law of thermodynamics can be expressed as
$$
d U=T d S-P d V
$$
By calculating and equating $\hat{\partial}^{2} U / \partial Y \partial X$ and $\partial^{2} U / \partial X \partial Y$, where $X$ and $Y$ are an unspecified pair of variables (drawn from $P, V, T$ and $S$ ), prove that
$$
\frac{\partial(S, T)}{\partial(X, Y)}=\frac{\partial(V, P)}{\partial(X, Y)}
$$
Using the properties of Jacobians, deduce that
$$
\frac{\partial(S, T)}{\partial(V, P)}=1
$$

Eduard Sanchez
Eduard Sanchez
Numerade Educator
10:10

Problem 23

This is a more difficult question about 'volumes' in an increasing number of dimensions.
(a) Let $R$ be a real positive number and define $K_{m}$ by
$$
K_{m}=\int_{-R}^{R}\left(R^{2}-x^{2}\right)^{m} d x
$$
Show, using integration by parts, that $K_{m}$ satisfies the recurrence relation
$$
(2 m+1) K_{m}=2 m R^{2} K_{m-1}
$$(b) For integer $n$, define $I_{n}=K_{n}$ and $J_{n}=K_{n+1 / 2}$. Evaluate $I_{0}$ and $J_{0}$ directly and hence prove that
$$
I_{n}=\frac{2^{2 n+1}(n !)^{2} R^{2 n+1}}{(2 n+1) !} \quad \text { and } \quad J_{n}=\frac{\pi(2 n+1) ! R^{2 n+2}}{2^{2 n+1} n !(n+1) !}
$$
(c) A sequence of functions $V_{n}(R)$ is defined by
$$
\begin{aligned}
&V_{0}(R)=1 \\
&V_{n}(R)=\int_{-R}^{R} V_{n-1}\left(\sqrt{R^{2}-x^{2}}\right) d x, \quad n \geq 1
\end{aligned}
$$
Prove by induction that
$$
V_{2 n}(R)=\frac{\pi^{n} R^{2 n}}{n !}, \quad V_{2 n+1}(R)=\frac{\pi^{n} 2^{2 n+1} n ! R^{2 n+1}}{(2 n+1) !}
$$
(d) For interest,
(i) show that $V_{2 n+2}(1)<V_{2 n}(1)$ and $V_{2 n+1}(1)<V_{2 n-1}(1)$ for all $n \geq 3$;
(ii) hence, by explicitly writing out $V_{k}(R)$ for $1 \leq k \leq 8$ (say), show that the "volume' of the totally symmetric solid of unit radius is a maximum in five dimensions.

WZ
Wen Zheng
Numerade Educator