Chapter 13

Evaluate the following integrals as they are written. $$\int_{0}^{1} \int_{-2 \sqrt{1-y^{2}}}^{2 \sqrt{1-y^{2}}} 2 x d x d y$$

Use a change of variables to evaluate the following integrals. \(\iiint_{D} z d V ; D\) is bounded by the paraboloid \(z=16-x^{2}-4 y^{2}\) and the \(x y\) -plane. Use \(x=4 u \cos v, y=2 u \sin v, z=w\)

Evaluate the following integrals in spherical coordinates. $$\int_{0}^{\pi} \int_{0}^{\pi / 6} \int_{2 \sec \varphi}^{4} \rho^{2} \sin \varphi d \rho d \varphi d \theta$$

Sketch each region and use a double integral to find its area. The region inside both the cardioid \(r=1+\sin \theta\) and the cardioid \(r=1+\cos \theta\)

Evaluate the following iterated integrals. $$\int_{1}^{2} \int_{1}^{2} \frac{x}{x+y} d y d x$$

Evaluate the following integrals as they are written. $$\int_{0}^{\ln 2} \int_{e^{y}}^{2} \frac{y}{x} d x d y$$

Use a change of variables to evaluate the following integrals. \(\iiint_{D} d V ; D\) is bounded by the upper half of the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1\) and the \(x y\) -plane. Use \(x=3 u\) \(y=2 v, z=w\)

Evaluate the following integrals in spherical coordinates. $$\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{1}^{2 \sec \varphi}\left(\rho^{-3}\right) \rho^{2} \sin \varphi d \rho d \varphi d \theta$$

Sketch each region and use a double integral to find its area. The region bounded by the spiral \(r=2 \theta,\) for \(0 \leq \theta \leq \pi,\) and the \(x\) -axis

Evaluate the following iterated integrals. $$\int_{0}^{2} \int_{0}^{1} x^{5} y^{2} e^{x^{3} y^{3}} d y d x$$

Evaluate the following integrals as they are written. $$\int_{0}^{4} \int_{y}^{2 y} x y d x d y$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If the transformation \(T: x=g(u, v), y=h(u, v)\) is linear in \(u\) and \(v,\) then the Jacobian is a constant. b. The transformation \(x=a u+b v, y=c u+d v\) generally maps triangular regions to triangular regions. c. The transformation \(x=2 v, y=-2 u\) maps circles to circles.

Evaluate the following integrals in spherical coordinates. $$\int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 3} \int_{0}^{2 \csc \varphi} \rho^{2} \sin \varphi d \rho d \varphi d \theta$$

Find the following average values. The average distance between points of the disk \(\\{(r, \theta): 0 \leq r \leq a\\}\) and the origin

Find the following average values. The average of the squared distance between the origin and points in the solid cylinder \(D=\left\\{(x, y, z): x^{2}+y^{2} \leq 4,0 \leq z \leq 2\right\\}\)

Evaluate the following iterated integrals. $$\int_{0}^{1} \int_{1}^{4} \frac{3 y}{\sqrt{x+y^{2}}} d x d y$$

Evaluate the following integrals as they are written. $$\int_{0}^{\pi / 2} \int_{y}^{\pi / 2} 6 \sin (2 x-3 y) d x d y$$

Use polar coordinates to find the centroid of the following constant-density plane regions. The quarter-circular disk \(R=\\{(r, \theta): 0 \leq r \leq 2,0 \leq \theta \leq \pi / 2\\}\)

Use spherical coordinates to find the volume of the following solids. A ball of radius \(a>0\)

Find the following average values. The average distance between points within the cardioid \(r=1+\cos \theta\) and the origin

Find the following average values. The average of the squared distance between the origin and points in the solid paraboloid \(D=\left\\{(x, y, z): 0 \leq z \leq 4-x^{2}-y^{2}\right\\}\)

Evaluate the following iterated integrals. $$\int_{1}^{4} \int_{0}^{2} e^{y \sqrt{x}} d y d x$$

Evaluate the following integrals as they are written. $$\int_{0}^{\pi / 2} \int_{0}^{\cos y} e^{\sin y} d x d y$$

Spherical coordinates Evaluate the Jacobian for the transformation from spherical to rectangular coordinates: \(x=\rho \sin \varphi \cos \theta, y=\rho \sin \varphi \sin \theta, z=\rho \cos \varphi .\) Show that \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\)

Use polar coordinates to find the centroid of the following constant-density plane regions. The region bounded by the cardioid \(r=1+\cos \theta\)

Use spherical coordinates to find the volume of the following solids. The solid bounded by the sphere \(\rho=2 \cos \varphi\) and the hemisphere \(\rho=1, z \geq 0\)

Find the following average values. The average \(z\) -coordinate of points on and within a hemisphere of radius 4 centered at the origin with its base in the \(x y\) -plane

Find the volume of the following solids. The solid beneath the cylinder \(f(x, y)=e^{-x}\) and above the region \(R=\\{(x, y): 0 \leq x \leq \ln 4,-2 \leq y \leq 2\\}\)

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} 12 y d A ; R\) is bounded by \(y=2-x, y=\sqrt{x},\) and \(y=0\).

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Find the area of \(R\)

Use polar coordinates to find the centroid of the following constant-density plane regions. The region bounded by the cardioid \(r=3-3 \cos \theta\)

Use spherical coordinates to find the volume of the following solids. The solid cardioid of revolution \(D=\\{(\rho, \varphi, \theta): 0 \leq \rho \leq 1+\cos \varphi, 0 \leq \varphi \leq \pi, 0 \leq \theta \leq 2 \pi\\}\)

Find the following average values. The average of the squared distance between the \(z\) -axis and points in the conical solid \(D=\\{(x, y, z): 2 \sqrt{x^{2}+y^{2}} \leq z \leq 8\\}\)

Find the volume of the following solids. The solid beneath the plane \(f(x, y)=6-x-2 y\) and above the region \(R=\\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 1\\}\)

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} y^{2} d A ; R\) is bounded by \(y=1, y=1-x,\) and \(y=x-1\).

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Evaluate \(\iint_{R}|x y| d A\)

Use polar coordinates to find the centroid of the following constant-density plane regions. The region bounded by one leaf of the rose \(r=\sin 2 \theta,\) for \(0 \leq \theta \leq \pi / 2\) \((\bar{x}, \bar{y})=\left(\frac{128}{105 \pi}, \frac{128}{105 \pi}\right)$$(\bar{x}, \bar{y})=\left(\frac{17}{18}, 0\right)\)

Determine whether the following statements are true and give an explanation or counterexample. a. An iterated integral of a function over the box \(D=\\{(x, y, z): 0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c\\}\) can be expressed in eight different ways. b. One possible iterated integral of \(f\) over the prism \(D=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 3 x-3,0 \leq z \leq 5\\}\) is \(\int_{0}^{3 x-3} \int_{0}^{1} \int_{0}^{5} f(x, y, z) d z d x d y\) \(\begin{aligned} &\text { c. The region } D=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq \sqrt{1-x^{2}}\\\ &0 \leq z \leq \sqrt{1-x^{2}}\\} \text { is a sphere. } \end{aligned}\)

Find the volume of the following solids. The solid beneath the plane \(f(x, y)=24-3 x-4 y\) and above the region \(R=\\{(x, y):-1 \leq x \leq 3,0 \leq y \leq 2\\}\) \(f(x, y)=24-3 x-4 y\)

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

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

Use polar coordinates to find the centroid of the following constant-density plane regions. The region bounded by the limaçon \(r=2+\cos \theta\)

Use spherical coordinates to find the volume of the following solids. The solid bounded by the cylinders \(r=1\) and \(r=2,\) and the cones \(\varphi=\pi / 6\) and \(\varphi=\pi / 3\)

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \sqrt{x^{2}+y^{2}} d y d x$$

Find the volume of the following solids. The solid beneath the paraboloid \(f(x, y)=12-x^{2}-2 y^{2}\) and above the region \(R=\\{(x, y): 1 \leq x \leq 2,0 \leq y \leq 1\\}\)

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

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a>0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\) Find the average square of the distance between points of \(R\) and the origin.

A thin (one-dimensional) wire of constant density is bent into the shape of a semicircle of radius \(a\). Find the location of its center of mass. (Hint: Treat the wire as a thin halfannulus with width \(\Delta a,\) and then let \(\Delta a \rightarrow 0\).)

Use spherical coordinates to find the volume of the following solids. That part of the ball \(\rho \leq 4\) that lies between the planes \(z=2\) and \(z=2 \sqrt{3}\)

Evaluate the following integrals using the method of your choice. A sketch is helpful. $$\int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}\left(x^{2}+y^{2}\right)^{3 / 2} d y d x$$