Chapter 13

Use a triple integral to compute the volume of the following regions. The parallelepiped (slanted box) with vertices (0,0,0),(1,0,0) \((0,1,0),(1,1,0),(0,1,1),(1,1,1),(0,2,1),\) and \((1,2,1) .\) (Use integration and find the best order of integration.)

Let \(R=\\{(x, y): 0 \leq x \leq \pi\) \(0 \leq y \leq a\\} .\) For what values of \(a,\) with \(0 \leq a \leq \pi,\) is \(\iint_{R} \sin (x+y) d A\) equal to \(1 ?\)

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} 3 x^{2} d A ; R\) is bounded by \(y=0, y=2 x+4,\) and \(y=x^{3}\).

A thin plate of unit density occupies the region between the parabola \(y=a x^{2}\) and the horizontal line \(y=b,\) where \(a>0\) and \(b>0 .\) Show that the center of mass is \(\left(0, \frac{3 b}{5}\right),\) independent of \(a\)

Use spherical coordinates to find the volume of the following solids. The solid inside the cone \(z=\left(x^{2}+y^{2}\right)^{1 / 2}\) that lies between the planes \(z=1\) and \(z=2\)

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\int_{-4}^{4} \int_{0}^{\sqrt{16-y^{2}}}\left(16-x^{2}-y^{2}\right) d x d y$$

Use a triple integral to compute the volume of the following regions. The larger of two solids formed when the parallelepiped (slanted box) with vertices (0,0,0),(2,0,0),(0,2,0),(2,2,0),(0,1,1) \((2,1,1),(0,3,1),\) and (2,3,1) is sliced by the plane \(y=2\)

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} x^{2} y d A ; R\) is bounded by \(y=0, y=\sqrt{x},\) and \(y=x-2\).

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Find the volume of \(D\)

Determine whether the following statements are true and give an explanation or counterexample. a. Any point on the \(z\) -axis has more than one representation in both cylindrical and spherical coordinates. b. The sets \(\\{(r, \theta, z): r=z\\}\) and \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) are the same.

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\int_{0}^{\pi / 4} \int_{0}^{\sec \theta} r^{3} d r d \theta$$

Use a triple integral to compute the volume of the following regions. The pyramid with vertices (0,0,0),(2,0,0),(2,2,0),(0,2,0) and (0,0,4)

Use double integrals to calculate the volume of the following regions. The tetrahedron bounded by the coordinate planes \((x=0, y=0, z=0)\) and the plane \(z=8-2 x-4 y\)

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Evaluate \(\iiint_{D}|x y z| d A\)

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A solid rectangular box has sides of length \(a, b,\) and \(c .\) Where is the center of mass relative to the faces of the box?

Spherical to rectangular Convert the equation \(\rho^{2}=\sec 2 \varphi\) where \(0 \leq \varphi<\pi / 4,\) to rectangular coordinates and identify the surface.

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\iint_{R} \sqrt{x^{2}+y^{2}} d A ; R=\\{(x, y): 0 \leq y \leq x \leq 1\\}$$

Use a triple integral to compute the volume of the following regions. The solid common to the cylinders \(z=\sin x\) and \(z=\sin y\) over the square \(R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}\) (The figure shows the cylinders, but not the common region.)

Use double integrals to calculate the volume of the following regions. The solid in the first octant bounded by the coordinate planes and the surface \(z=1-y-x^{2}\)

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Find the center of mass of the upper half of \(D(z \geq 0)\) assuming it has a constant density.

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A solid cone has a base with a radius of \(a\) and a height of \(h\). How far from the base is the center of mass?

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\iint_{R} \sqrt{x^{2}+y^{2}} d A ; R=\left\\{(x, y): 1 \leq x^{2}+y^{2} \leq 4\right\\}$$

Use a triple integral to compute the volume of the following regions. The wedge of the square column \(|x|+|y|=1\) created by the planes \(z=0\) and \(x+y+z=1\)

Suppose a thin rectangular plate, represented by a region \(R\) in the \(x y\) -plane, has a density given by the function \(\rho(x, y) ;\) this function gives the area density in units such as grams per square centimeter \(\left(\mathrm{g} / \mathrm{cm}^{2}\right)\) The mass of the plate is \(\iint_{R} \rho(x, y) d A .\) Assume that \(R=\\{(x, y): 0 \leq x \leq \pi / 2,0 \leq y \leq \pi\\}\) and find the mass of the plates with the following density functions. a. \(\rho(x, y)=1+\sin x\) b. \(\rho(x, y)=1+\sin y\) c. \(\overline{\rho(x, y)}=1+\sin x \sin y\)

Use double integrals to calculate the volume of the following regions. The segment of the cylinder \(x^{2}+y^{2}=1\) bounded above by the plane \(z=12+x+y\) and below by \(z=0\)

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The ball of radius 4 centered at the origin with a density \(f(\rho, \varphi, \theta)=1+\rho\)

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Find the average square of the distance between points of \(D\) and the origin.

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A solid is enclosed by a hemisphere of radius \(a\). How far from the base is the center of mass?

Evaluate the following integrals using the method of your choice. A sketch is helpful. \(\iint_{R} \frac{x-y}{x^{2}+y^{2}+1} d A ; R\) is the region bounded by the unit circle centered at the origin.

Consider the region \(D_{1}=\\{(x, y, z): 0 \leq x \leq y \leq z \leq 1\\}\) a. Find the volume of \(D_{1}\). b. Let \(D_{2}, \ldots ., D_{6}\) be the "cousins" of \(D_{1}\) formed by rearranging \(x, y,\) and \(z\) in the inequality \(0 \leq x \leq y \leq z \leq 1 .\) Show that the volumes of \(D_{1}, \ldots, D_{6}\) are equal. c. Show that the union of \(D_{1}, \ldots, D_{6}\) is a unit cube.

Use double integrals to calculate the volume of the following regions. The solid beneath the cylinder \(z=y^{2}\) and above the region \(R=\\{(x, y): 0 \leq y \leq 1, y \leq x \leq 1\\}\)

Parabolic coordinates Let \(T\) be the transformation \(x=u^{2}-v^{2}\) \(y=2 u v\) a. Show that the lines \(u=a\) in the \(u v\) -plane map to parabolas in the \(x y\) -plane that open in the negative \(x\) -direction with vertices on the positive \(x\) -axis. b. Show that the lines \(v=b\) in the \(u v\) -plane map to parabolas in the \(x y\) -plane that open in the positive \(x\) -direction with vertices on the negative \(x\) -axis. c. Evaluate \(J(u, v)\) d. Use a change of variables to find the area of the region bounded by \(x=4-y^{2} / 16\) and \(x=y^{2} / 4-1\) e. Use a change of variables to find the area of the curved rectangle above the \(x\) -axis bounded by \(x=4-y^{2} / 16\) \(x=9-y^{2} / 36, x=y^{2} / 4-1,\) and \(x=y^{2} / 64-16\) f. Describe the effect of the transformation \(x=2 u v\) \(y=u^{2}-v^{2}\) on horizontal and vertical lines in the \(u v\) -plane.

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The ball of radius 8 centered at the origin with a density \(f(\rho, \varphi, \theta)=2 e^{-\rho^{3}}\)

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A region is enclosed by an isosceles triangle with two sides of length \(s\) and a base of length \(b\). How far from the base is the center of mass?

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\begin{array}{l} \iint_{R} \frac{d A}{4+\sqrt{x^{2}+y^{2}}} ; R=\\{(r, \theta): 0 \leq r \leq 2 \\\ \pi / 2 \leq \theta \leq 3 \pi / 2\\} \end{array}$$

Write the integral \(\int_{0}^{2} \int_{0}^{1} \int_{0}^{1-y} d z d y d x\) in the five other possible orders of integration.

Let \(S\) be the solid in \(\mathbb{R}^{3}\) between the cylinder \(z=f(x)\) and the region \(R=\\{(x, y): a \leq x \leq b, c \leq y \leq d\\},\) where \(f(x) \geq 0\) on \(R .\) Explain why \(\int_{c}^{d} \int_{a}^{b} f(x) d x d y\) equals the area of the constant cross section of \(S\) multiplied by \((d-c),\) which is the volume of \(S\)

Reverse the order of integration in the following integrals. $$\int_{0}^{2} \int_{x^{2}}^{2 x} f(x, y) d y d x$$

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The solid cone \(\\{(r, \theta, z): 0 \leq z \leq 4,0 \leq r \leq \sqrt{3} z\) \(0 \leq \theta \leq 2 \pi\\}\) with a density \(f(r, \theta, z)=5-z\)

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A tetrahedron is bounded by the coordinate planes and the plane \(x / a+y / a+z / a=1 .\) What are the coordinates of the center of mass?

Use integration to show that the circles \(r=2 a \cos \theta\) and \(r=2 a \sin \theta\) have the same area, which is \(\pi a^{2}\)

Let \(D\) be the solid bounded by \(y=x, z=1-y^{2}, x=0\) and \(z=0 .\) Write triple integrals over \(D\) in all six possible orders of integration.

Reverse the order of integration in the following integrals. $$\int_{0}^{3} \int_{0}^{6-2 x} f(x, y) d y d x$$

Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function \(f\) to avoid confusion with the radial spherical coordinate \(\rho\). The solid cylinder \(\\{(r, \theta, z): 0 \leq r \leq 2,0 \leq \theta \leq 2 \pi\) \(-1 \leq z \leq 1\\}\) with a density of \(f(r, \theta, z)=(2-|z|)(4-r)\)

Consider the following two-and three-dimensional regions. Specify the surfaces and curves that bound the region, choose a convenient coordinate system, and compute the center of mass assuming constant density. All parameters are positive real numbers. A tetrahedron is bounded by the coordinate planes and the plane \(x / a+y / a+z / a=1 .\) What are the coordinates of the center of mass?

Which bowl holds more water if it is filled to a depth of 4 units? \(\cdot\) The paraboloid \(z=x^{2}+y^{2},\) for \(0 \leq z \leq 4\) \(\cdot\) The cone \(z=\sqrt{x^{2}+y^{2}},\) for \(0 \leq z \leq 4\) \(\cdot\) The hyperboloid \(z=\sqrt{1+x^{2}+y^{2}},\) for \(1 \leq z \leq 5\)

Reverse the order of integration in the following integrals. $$\int_{1 / 2}^{1} \int_{0}^{-\ln y} f(x, y) d x d y$$

Linear transformations Consider the linear transformation \(T\) in \(\mathbb{R}^{2}\) given by \(x=a u+b v, y=c u+d v,\) where \(a, b, c,\) and \(d\) are real numbers, with \(a d \neq b c\) a. Find the Jacobian of \(T\) b. Let \(S\) be the square in the \(u v\) -plane with vertices (0,0) \((1,0),(0,1),\) and \((1,1),\) and let \(R=T(S) .\) Show that \(\operatorname{area}(R)=|J(u, v)|\) c. Let \(\ell\) be the line segment joining the points \(P\) and \(Q\) in the uv- plane. Show that \(T(\ell)\) (the image of \(\ell\) under \(T\) ) is the line segment joining \(T(P)\) and \(T(Q)\) in the \(x y\) -plane. (Hint: Use vectors.) d. Show that if \(S\) is a parallelogram in the \(u v\) -plane and \(R=T(S),\) then \(\operatorname{area}(R)=|J(u, v)| \operatorname{area}(S) .\) (Hint: Without loss of generality, assume the vertices of \(S\) are \((0,0),(A, 0)\) \((B, C),\) and \((A+B, C),\) where \(A, B,\) and \(C\) are positive, and use vectors.)

Changing order of integration If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration. The solid outside the cylinder \(r=1\) and inside the sphere \(\rho=5\) for \(z \geq 0,\) in the orders \(d z d r d \theta, d r d z d \theta,\) and \(d \theta d z d r\)

Suppose a wedge of cheese fills the region in the first octant bounded by the planes \(y=z, y=4,\) and \(x=4\) You could divide the wedge into two pieces of equal volume by slicing the wedge with the plane \(x=2 .\) Instead find \(a\) with \(0