Question 6.102

• (a) Find a linear transformation taking the square \(S=[0,1]\times [0,1]\) to the parallelogram \(P\) with vertices (0, 0), (2, 0), (1, 2), (3, 2).
• (b) Write down a change of variables formula appropriate to the transformation you found in part (a).

Question 6.103

• (a) Find the image of the square \([0,1]\times [0,1]\) under the transformation \(T(x,y)= (2x, x+3y)\).
• (b) Write down a change of variables formula appropriate to the transformation and the region you found in part (a).

Question 6.104

Let \(B\) be the region in the first quadrant bounded by the curves \(xy=1,xy=3, x^2-y^2=1\), and \(x^2-y^2=4\). Evaluate \({\intop\!\!\intop}_B(x^2+y^2)\,{\it dx}\,{\it dy}\) using the change of variables \(u=x^2-y^2\), \(v=xy\).

Question 6.105

In parts (a) to (d), make the indicated change of variables. (Do not evaluate.)

• (a) \( {\displaystyle \int\nolimits^1_0 \int\nolimits^1_{-1}\int\nolimits^{\sqrt{(1-y^2)}}_{-\sqrt{(1-y^2)}}}\ (x^2+y^2)^{1/2} {\it dx}\,{\it dy}\,{\it dz}\), cylindrical coordinates
• (b) \({\displaystyle \int\nolimits^{1}_{-1}\int\nolimits^{\sqrt{(1-y^2)}}_{-\sqrt{(1-y^2)}} \int\nolimits^{\sqrt{(4-x^2-y^2)}}_{-\sqrt{(4-x^2-y^2)}}}\ xyz\, {\it dz}\,{\it dx}\,{\it dy}\), cylindrical coordinates
• (c) \({ \displaystyle\int\nolimits^{\sqrt{2}}_{-\sqrt{2}}\int\nolimits^{\sqrt{(2-y^2)}}_{-\sqrt{(2-y^2)}} \int\nolimits^{\sqrt{(4-x^2-y^2)}}_{\sqrt{(x^2+y^2)}}}\ z^2{\it dz}\,{\it dx}\,{\it dy}\), spherical coordinates
• (d) \( {\displaystyle\int\nolimits^1_0\int\nolimits_0^{\pi/4} \int\nolimits^{2\pi}_0}\rho^3\sin \,2\phi\, {\,d} \theta \,{\,d} \phi\, {\,d} \rho\), rectangular coordinates

Question 6.106

Find the volume inside the surfaces \(x^2+y^2=z\) and \(x^2+y^2+z^2=2\).

Question 6.107

Find the volume enclosed by the cone \(x^2+y^2=z^2\) and the plane \(2z-y-2=0\).

Question 6.108

A cylindrical hole of diameter 1 is bored through a sphere of radius 2. Assuming that the axis of the cylinder passes through the center of the sphere, find the volume of the solid that remains.

Question 6.109

Let \(C_1\) and \(C_2\) be two cylinders of infinite extent, of diameter 2, and with axes on the \(x\) and \(y\) axes, respectively. Find the volume of their intersection, \(C_1\cap C_2\).

Question 6.110

Find the volume bounded by \(x/a+y/b+z/c=1\) and the coordinate planes.

Question 6.111

Find the volume determined by \(z\leq 6-x^2-y^2\) and \(z\geq \sqrt{x^2+y^2}\).

Question 6.112

The tetrahedron defined by \(x\geq 0,y\geq 0,z\geq 0,x+y+z\leq 1\) is to be sliced into \(n\) segments of equal volume by planes parallel to the plane \(x+y+z=1\). Where should the slices be made?

Question 6.113

Let \(E\) be the solid ellipsoid \(E=\{(x,y,z)\mid (x^2/a^2)+\) \((y^2/b^2)+(z^2/c^2)\leq 1\}\), where \(a > 0,b > 0\), and \(c > 0\). Evaluate \[ {\intop\!\!\!\intop\!\!\!\intop} xyz\,{\it dx}\,{\it dy}\,{\it dz} \]

• (a) over the whole ellipsoid; and
• (b) over that part of it in the first quadrant: \[ x\geq 0,\quad y\geq 0,\quad \hbox{ and }\quad z\geq 0,\quad \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}\leq 1. \]

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Question 6.114

Find the volume of the “ice cream cone” defined by the inequalities \(x^2+y^2\leq \textstyle \frac{1}{5}z^2,\) and \(0\leq z\leq\) \(5+\sqrt{5-x^2-y^2}\).

Question 6.115

Let \(\rho,\theta,\phi\) be spherical coordinates in \({\mathbb R}^3\) and suppose that a surface surrounding the origin is described by a continuous positive function \(\rho=f(\theta,\phi)\). Show that the volume enclosed by the surface is \[ V=\frac{1}{3}\int^{2\pi}_0\int^\pi_0[f(\theta,\phi)]^3\sin \phi {\,d} \phi {\,d} \theta. \]

Question 6.116

Using an appropriate change of variables, evaluate \[ \int\!\!\!\int_B \exp \,[(y-x)/(y+x)]\,{\it dx}\,{\it dy}, \] where \(B\) is the interior of the triangle with vertices at \((0, 0), (0, 1)\), and \((1, 0)\).

Question 6.117

Suppose the density of a solid of radius \(R\) is given by \((1+d^3)^{-1}\), where \(d\) is the distance to the center of the sphere. Find the total mass of the sphere.

Question 6.118

The density of the material of a spherical shell whose inner radius is 1 m and whose outer radius is 2 m is \(0.4 d^2\) g/cm\(^3\), where \(d\) is the distance to the center of the sphere in meters. Find the total mass of the shell.

Question 6.119

If the shell in Exercise 17 were dropped into a large tank of pure water, would it float? What if the shell leaked? (Assume that the density of water is exactly 1 g/cm\(^3\).)

Question 6.120

The temperature at points in the cube \(C=\{(x,y,z)\mid -\) \(1\leq x\leq 1,-1\leq y\leq 1\), and \(-1\leq z\leq 1\}\) is 32\(d^2\), where \(d\) is the distance to the origin.

• (a) What is the average temperature?
• (b) At what points of the cube is the temperature equal to the average temperature?

Question 6.121

Use cylindrical coordinates to find the center of mass of the region defined by \[ y^2+z^2\leq \frac{1}{4},\qquad (x-1)^2+y^2+z^2\leq 1, \qquad x\geq 1. \]

Question 6.122

Find the center of mass of the solid hemisphere \[ V=\{(x,y,z)\mid x^2+y^2+z^2\leq a^2 \hbox{ and }z\geq 0\}, \] if the density is constant.

Question 6.123

Evaluate \({\intop\!\!\intop}_B e^{-x^2-y^2}{\it dx}\,{\it dy}\), where \(B\) consists of those \((x,y)\) satisfying \(x^2+y^2\leq 1\) and \(y\leq 0\).

Question 6.124

Evaluate \[ \intop\!\!\!\intop\!\!\!\intop\nolimits_{\! S}\, \frac{{\it dx}\,{\it dy}\,{\it dz}}{(x^2+y^2+z^2)^{3/2}}, \] where \(S\) is the solid bounded by the spheres \(x^2+y^2+z^2=a^2\) and \(x^2+y^2+z^2=b^2\), where \(a>b>0\).

Question 6.125

Evaluate \(\displaystyle\int\!\!\!\int\!\!\!\int_D(x^2+y^2+z^2)xyz\, {\it dx}\,{\it dy}\,{\it dz}\) over each of the following regions.

• (a) The sphere \(D=\{(x,y,z)\mid x^2+y^2+z^2\leq R^2\}\)
• (b) The hemisphere \(D=\{(x,y,z)\mid x^2+\) \(y^2+z^2\leq R^2 \hbox{ and } z\geq 0\}\)
• (c) The octant \(D=\{(x,y,z)\mid x\geq 0,y\geq 0,z\geq 0\), and \(z^2+y^2+z^2\leq R^2\}\)

Question 6.126

Let \(C\) be the cone-shaped region \(\{(x,y,z)\mid \sqrt{x^2+y^2} \leq z\leq 1\}\) in \({\mathbb R}^3\) and evaluate the integral \(\displaystyle\int\!\!\!\int\!\!\!\int_C(1+\sqrt{x^2+y^2})\,{\it dx}\,{\it dy}\,{\it dz}\).

Question 6.127

Find \(\displaystyle\int\!\!\!\int\!\!\!\int_{{\mathbb R}^3}f(x,y,z)\,{\it dx}\,{\it dy}\,{\it dz}\), where \(f(x,y,z)=\exp\ [-(x^2+y^2+z^2)^{3/2}]\).

Question 6.128

The flexural rigidity EI of a uniform beam is the product of its Young’s modulus of elasticity \(E\) and the moment of inertia \(I\) of the cross section of the beam with respect to a horizontal line \(l\) passing through the center of gravity of this cross section. Here \[ I= \intop\!\!\!\intop\nolimits_{R}\ [d(x,y)]^2{\it dx}\,{\it dy}, \] where \(d(x,y)=\) the distance from \((x,y)\) to \(l\) and \(R=\) the cross section of the beam being considered.

• (a) Assume that the cross section \(R\) is the rectangle \(-1\leq x\leq 1,-1\leq y\leq 2\), and \(l\) is the line \(y=1/2\). Find \(I\).
• (b) Assume the cross section \(R\) is a circle of radius 4 and \(l\) is the \(x\) axis. Find \(I\), using polar coordinates.

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Question 6.129

Find, \({\intop\!\!\intop\!\!\intop}_{{\mathbb R}^3}f(x,y,z)\,{\it dx}\,{\it dy}\,{\it dz}\), where \[ f(x,y,z)=\frac{1}{[1+(x^2+y^2+z^2)^{3/2}]^{3/2}}. \]

Question 6.130

Suppose \(D\) is the unbounded region of \({\mathbb R}^2\) given by the set of \((x,y)\) with \(0\leq x < \infty, 0\leq y\leq x\). Let \(f(x,y)=x^{-3/2}e^{y-x}\). Does the improper integral \({\intop\!\!\intop}_Df(x,y)\,{\it dx}\,{\it dy}\) exist?

Question 6.131

If the world were two-dimensional, the laws of physics would predict that the gravitational potential of a mass point is proportional to the logarithm of the distance from the point. Using polar coordinates, write an integral giving the gravitational potential of a disk of constant density.

Question 6.132

• (a) Evaluate the improper integral \[ \int^{\infty}_0\int^y_0 xe^{-y^3}{\it dx}\,{\it dy}. \]
• (b) Evaluate \[ \int\!\!\!\int_B(x^4+2x^2y^2+y^4)\,{\it dx}\,{\it dy}, \] where \(B\) is the portion of the disk of radius 2 [centered at \((0, 0)\) in the first quadrant].

Question 6.133

Let \(f\) be a nonnegative function on an \(x\)-simple or a \(y\)-simple region \(D\subset {\mathbb R}^2\) and that is continuous except for points on the boundary of \(D\) and at most finitely many points interior to \(D\). Give a suitable definition of \({\intop\!\!\intop}_D f{\it {\,d} A}\).

Question 6.134

Evaluate \({\intop\!\!\intop}_{{\mathbb R}^2}f(x,y)\,{\it dx}\,{\it dy}\), where \(f(x,y)=\) \(1/(1+x^2+y^2)^{3/2}\). (HINT: You may assume that changing variables and Fubini’s theorem are valid for improper integrals.)