Chapter 13

Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, \(R\) and \(S\) \(\iint_{R} e^{x y} d A,\) where \(R\) is the region bounded by the hyperbolas \(x y=1\) and \(x y=4,\) and the lines \(y / x=1\) and \(y / x=3\)

Sketch the following regions \(R\). Then express \(\iint_{R} f(r, \theta) d A\) as an iterated integral over \(R\). The region inside the leaf of the rose \(r=2 \sin 2 \theta\) in the first quadrant

Evaluate the following integrals. $$\int_{0}^{1} \int_{y}^{2-y} \int_{0}^{2-x-y} x y d z d x d y$$

Average value Compute the average value of the following functions over the region \(R\). $$f(x, y)=\sin x \sin y ; R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}$$

Find the coordinates of the center of mass of the following solids with variable density. The solid bounded by the upper half of the sphere \(\rho=6\) and \(z=0\) with density \(f(\rho, \varphi, \theta)=1+\rho / 4\)

Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, \(R\) and \(S\) \(\iint_{R} x y d A,\) where \(R\) is the region bounded by the hyperbolas \(-x y=1\) and \(x y=4,\) and the lines \(y=1\) and \(y=3\)

Identify and sketch the following sets in spherical coordinates. $$\\{(\rho, \varphi, \theta): 1 \leq \rho \leq 3\\}$$

Average value Compute the average value of the following functions over the region \(R\). $$f(x, y)=\sin x \sin y ; R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d x d y\). The region bounded by \(y=4-x, y=1,\) and \(x=0\)

Find the coordinates of the center of mass of the following solids with variable density. The interior of the cube in the first octant formed by the planes \(x=1, y=1,\) and \(z=1\) with \(\rho(x, y, z)=2+x+y+z\)

Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, \(R\) and \(S\) \(\iint_{R}(x-y) \sqrt{x-2 y} d A,\) where \(R\) is the triangular region bounded by \(y=0, x-2 y=0,\) and \(x-y=1\)

Identify and sketch the following sets in spherical coordinates. $$\\{(\rho, \varphi, \theta): \rho=2 \csc \varphi, 0<\varphi<\pi\\}$$

Find the volume of the following solids. The solid bounded by \(x=0, x=2, y=0, y=e^{-z}, z=0,\) and \(z=1\)

Sketch the following regions \(R\). Then express \(\iint_{R} f(r, \theta) d A\) as an iterated integral over \(R\). The region outside the circle \(r=2\) and inside the circle \(r=4 \sin \theta\)

Find the average squared distance between the points of \(R=\\{(x, y):-2 \leq x \leq 2,0 \leq y \leq 2\\}\) and the origin.

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d x d y\). The region in quadrants 2 and 3 bounded by the semicircle with radius 3 centered at (0, 0)

Find the coordinates of the center of mass of the following solids with variable density. The interior of the prism formed by \(z=x, x=1, y=4,\) and the coordinate planes with \(\rho(x, y, z)=2+y\)

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$x=v+w, y=u+w, z=u+v$$

Identify and sketch the following sets in spherical coordinates. $$\\{(\rho, \varphi, \theta): \rho=4 \cos \varphi, 0 \leq \varphi \leq \pi / 2\\}$$

Sketch the following regions \(R\). Then express \(\iint_{R} f(r, \theta) d A\) as an iterated integral over \(R\). The region outside the circle \(r=1\) and inside the rose \(r=2 \sin 3 \theta\) in the first quadrant

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The region of integration for \(\int_{4}^{6} \int_{1}^{3} 4 d x d y\) is a square. b. If \(f\) is continuous on \(\mathbb{R}^{2}\), then $$ \int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{4}^{6} \int_{1}^{3} f(x, y) d y d x $$ c. If \(f\) is continuous on \(\mathbb{R}^{2}\), then $$ \int_{4}^{6} \int_{1}^{3} f(x, y) d x d y=\int_{1}^{3} \int_{4}^{6} f(x, y) d y d x $$

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$x=u+v-w, y=u-v+w, z=-u+v+w$$

Identify and sketch the following sets in spherical coordinates. $$\\{(\rho, \varphi, \theta): \rho=2 \sec \varphi, 0 \leq \varphi<\pi / 2\\}$$

Sketch the following regions \(R\). Then express \(\iint_{R} f(r, \theta) d A\) as an iterated integral over \(R\). The region outside the circle \(r=\frac{1}{2}\) and inside the cardioid \(r=1+\cos \theta\)

Find the volume of the following solids. The solid bounded by \(x=0, y=z^{2}, z=0,\) and \(z=2-x-y\)

Symmetry Evaluate the following integrals using symmetry arguments. Let \(R=\\{(x, y):-a \leq x \leq a,-b \leq y \leq b\\},\) where \(a\) and \(b\) are positive real numbers. a. \(\iint_{R} x y e^{-\left(x^{2}+y^{2}\right)} d A\) b. \(\iint_{R} \frac{\sin (x-y)}{x^{2}+y^{2}+1} d A\)

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d x d y\). The region in the first quadrant bounded by the \(x\) -axis, the line \(x=6-y,\) and the curve \(y=\sqrt{x}\)

Explain why or why not ,Determine whether the following statements are true and give an explanation or counterexample. a. A thin plate of constant density that is symmetric about the \(x\) -axis has a center of mass with an \(x\) -coordinate of zero. b. A thin plate of constant density that is symmetric about both the \(x\) -axis and the \(y\) -axis has its center of mass at the origin. c. The center of mass of a thin plate must lie on the plate. d. The center of mass of a connected solid region (all in one piece) must lie within the region.

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$x=v w, y=u w, z=u^{2}-v^{2}$$

Sketch each region and use a double integral to find its area. $$\text { The annular region }\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi\\}$$

Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. $$\int_{0}^{5} \int_{-1}^{0} \int_{0}^{4 x+4} d y d x d z \text { in the order } d z d x d y$$

Evaluate the following integrals as they are written. $$\int_{-1}^{2} \int_{y}^{4-y} d x d y$$

\(A\) thin rod of length \(L\) has a linear density given by \(\rho(x)=2 e^{-x / 3}\) on the interval \(0 \leq x \leq L\). Find the mass and center of mass of the rod. How does the center of mass change as \(L \rightarrow \infty ?\)

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$u=x-y, v=x-z, w=y+z \quad \text { (Solve for } x, y, \text { and } z \text { first.) }$$

Evaluate the following integrals in spherical coordinates. $$\iiint_{D} e^{-\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} d V ; D \text { is the unit ball. }$$

Sketch each region and use a double integral to find its area. The region bounded by the cardioid \(r=2(1-\sin \theta)\)

Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. $$\int_{0}^{1} \int_{-2}^{2} \int_{0}^{\sqrt{4-y^{2}}} d z d y d x \text { in the order } d y d z d x$$

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

Limiting center of mass \(A\) thin rod of length \(L\) has a linear density given by \(\rho(x)=\frac{10}{1+x^{2}}\) on the interval \(0 \leq x \leq L\). Find the mass and center of mass of the rod. How does the center of mass change as \(L \rightarrow \infty ?\)

Use a change of variables to evaluate the following integrals. $$\begin{aligned} &\iiint_{D} x y d V ; D \text { is bounded by the planes } y-x=0\\\ &y-x=2, z-y=0, z-y=1, z=0, \text { and } z=3 \end{aligned}$$

Evaluate the following integrals in spherical coordinates. \(\iiint_{D} \frac{d V}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} ; D\) is the solid between the spheres of radius 1 and 2 centered at the origin.

Sketch each region and use a double integral to find its area. The region bounded by all leaves of the rose \(r=2 \cos 3 \theta\)

Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{\sqrt{1-x^{2}}} d y d z d x \text { in the order } d z d y d x$$

Draw the solid whose volume is given by the following iterated integrals. Then find the volume of the solid. $$\int_{0}^{6} \int_{1}^{2} 10 d y d x$$

Evaluate the following integrals as they are written. $$\int_{0}^{4} \int_{-\sqrt{16-y^{2}}}^{\sqrt{16-y^{2}}} 2 x y d x d y$$

\(A\) thin plate is bounded by the graphs of \(y=e^{-x}, y=-e^{-x}, x=0,\) and \(x=L .\) Find its center of mass. How does the center of mass change as \(L \rightarrow \infty ?\)

Use a change of variables to evaluate the following integrals. \(\iiint_{D} d V ; D\) is bounded by the planes \(y-2 x=0, y-2 x=1\) \(z-3 y=0, z-3 y=1, z-4 x=0,\) and \(z-4 x=3\)

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

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 \(r=1+\cos \theta\)

Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. $$\int_{0}^{4} \int_{0}^{\sqrt{16-x^{2}}} \int_{0}^{\sqrt{16-x^{2}-z^{2}}} d y d z d x \text { in the order } d x d y d z$$