Chapter 13

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ S=\left\\{(x, y, z): 0 \leq x \leq \sqrt{y}, 0 \leq y \leq 4,0 \leq z \leq \frac{3}{2} x\right\\} $$

An iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the region. \(\int_{0}^{2 \pi} \int_{1}^{3} r d r d \theta\)

$$ \int_{\pi / 6}^{\pi / 2} \int_{0}^{\sin \theta} 6 r \cos \theta d r d \theta $$

Evaluate each of the iterated integrals. $$ \int_{0}^{1} \int_{0}^{2} \frac{y}{1+x^{2}} d y d x $$

In Problems 15-22, use spherical coordinates to find the indicated quantity. Mass of the solid inside the sphere \(\rho=b\) and outside the sphere \(\rho=a(a

An iterated integral is given either in rectangular or polar coordinates. The double integral gives the mass of some lamina \(R\). Sketch the lamina \(R\) and determine the density \(\delta .\) Then find the mass and center of mass. \(\int_{0}^{2} \int_{0}^{x} k d y d x\)

Find the transformation from the \(u v\) -plane to the xy-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x y, v=x $$

Find the area of the indicated surface. Make a sketch in each case. The part of \(z=9-x^{2}-y^{2}\) above the plane \(z=5\).

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{aligned} S=\\{(&x, y, z): 0 \leq x \leq 3 z \\ &0 \leq y \leq 4-x-2 z, 0 \leq z \leq 2\\} \end{aligned} $$

An iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the region. \(\int_{0}^{\pi / 2} \int_{0}^{\theta} r d r d \theta\)

In Problems \(15-20\), evaluate the given double integral by changing it to an iterated integral. $$ \iint_{S} x y d A ; S \text { is the region bounded by } y=x^{2} \text { and } y=1 \text { . } $$

Evaluate each of the iterated integrals. $$ \int_{0}^{\pi} \int_{0}^{3} y \cos ^{2} x d y d x $$

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R} 3 d A $$

Use spherical coordinates to find the indicated quantity. Mass of a solid inside a sphere of radius \(2 a\) and outside a circular cylinder of radius \(a\) whose axis is a diameter of the sphere, if the density is proportional to the square of the distance from the center of the sphere

Find the transformation from the \(u v\) -plane to the xy-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x^{2}, v=x y $$

Find the area of the indicated surface. Make a sketch in each case. The part of \(z=9-x^{2}\) above the \(x y\) -plane with \(0 \leq x \leq 20\).

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ S=\left\\{(x, y, z): 0 \leq x \leq y^{2}, 0 \leq y \leq \sqrt{z}, 0 \leq z \leq 1\right\\} $$

An iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the region. \(\int_{0}^{\pi / 2} \int_{0}^{\cos \theta} r d r d \theta\)

Evaluate the given double integral by changing it to an iterated integral. \(\iint_{S}(x+y) d A ; S\) is the triangular region with vertices \((0,0),(0,4)\), and \((1,4)\).

Evaluate each of the iterated integrals. $$ \int_{-1}^{1} \int_{0}^{1} x e^{x^{2}} d x d y $$

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}(x+1) d A $$

Use spherical coordinates to find the indicated quantity. Center of mass of a solid hemisphere of radius \(a\), if the density is proportional to the distance from the center of the sphere

An iterated integral is given either in rectangular or polar coordinates. The double integral gives the mass of some lamina \(R\). Sketch the lamina \(R\) and determine the density \(\delta .\) Then find the mass and center of mass. \(\int_{-3}^{3} \int_{0}^{9-x^{2}} k\left(x^{2}+y^{2}\right) d y d x\)

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ \(S\) is the tetrahedron with vertices \((0,0,0),(3,2,0),(0,3,0)\), and \((0,0,2)\).

An iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the region. \(\int_{0}^{\pi} \int_{0}^{\sin \theta} r d r d \theta\)

Evaluate the given double integral by changing it to an iterated integral. \(\iint_{S}\left(x^{2}+2 y\right) d A ; S\) is the region between \(y=x^{2}\) and \(y=\sqrt{x} .\)

Evaluate the indicated double integral over \(R\). $$ \iint_{R} x y^{3} d A ; R=\\{(x, y): 0 \leq x \leq 1,-1 \leq y \leq 1\\} $$

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}(y+1) d A $$

Use spherical coordinates to find the indicated quantity. Center of mass of a solid hemisphere of radius \(a\), if the density is proportional to the distance from the axis of symmetry

Use a transformation to evaluate the given double integral over the region \(R\) which is the triangle with vertices \((1,0),(4,0)\), and \((4,3) .\) $$ \iint_{R} \sqrt{\frac{x+y}{x-y}} d A $$

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ \(S\) is the region in the first octant bounded by the surface \(z=9-x^{2}-y^{2}\) and the coordinate planes.

An iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the region. \(\int_{0}^{3 \pi / 2} \int_{0}^{\theta^{2}} r d r d \theta\)

Evaluate the given double integral by changing it to an iterated integral. \(\iint_{S}\left(x^{2}-x y\right) d A ; S\) is the region between \(y=x\) and \(y=3 x-x^{2}\)

Evaluate the indicated double integral over \(R\). $$ \iint_{R}\left(x^{2}+y^{2}\right) d A ; R=\\{(x, y):-1 \leq x \leq 1,0 \leq y \leq 2\\} $$

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}(x-y+4) d A $$

Use a transformation to evaluate the given double integral over the region \(R\) which is the triangle with vertices \((1,0),(4,0)\), and \((4,3) .\) $$ \iint \sin (\pi(2 x-y)) \cos (\pi(y-2 x)) d A $$

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ \(S\) is the region in the first octant bounded by the cylinder \(y^{2}+z^{2}=1\) and the planes \(x=1\) and \(x=4 .\)

Evaluate by using polar coordinates. Sketch the region of integration first. \(\iint_{S} e^{x^{2}+y^{2}} d A\), where \(S\) is the region enclosed by \(x^{2}+y^{2}=4\)

Evaluate the given double integral by changing it to an iterated integral. \(\iint_{S} \frac{2}{1+x^{2}} d A ; S\) is the triangular region with vertices at \((0,0),(2,2)\), and \((0,2)\).

Evaluate the indicated double integral over \(R\). $$ \begin{array}{l} \iint_{R} \sin (x+y) d A \\ R=\\{(x, y): 0 \leq x \leq \pi / 2,0 \leq y \leq \pi / 2\\} \end{array} $$

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}\left(x^{2}+y^{2}\right) d A $$

Use spherical coordinates to find the indicated quantity. Volume of the solid within the sphere \(x^{2}+y^{2}+z^{2}=16\), outside the cone \(z=\sqrt{x^{2}+y^{2}}\), and above the \(x y\) -plane

An iterated integral is given either in rectangular or polar coordinates. The double integral gives the mass of some lamina \(R\). Sketch the lamina \(R\) and determine the density \(\delta .\) Then find the mass and center of mass. \(\int_{0}^{\pi / 2} \int_{0}^{\theta} k r d r d \theta\)

Use a transformation to evaluate the given double integral over the region \(R\) which is the triangle with vertices \((1,0),(4,0)\), and \((4,3) .\) $$ \iint_{R}(2 x-y) \cos (y-2 x) d A $$

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ \(S\) is the smaller region bounded by the cylinder \(x^{2}+y^{2}-2 y=0\) and the planes \(x-y=0, z=0\), and \(z=3\).

Evaluate by using polar coordinates. Sketch the region of integration first. \(\iint_{S} \sqrt{4-x^{2}-y^{2}} d A\), where \(S\) is the first quadrant sector of the circle \(x^{2}+y^{2}=4\) between \(y=0\) and \(y=x\)

Evaluate the indicated double integral over \(R\). $$ \begin{array}{l} \iint_{R} x y \sqrt{1+x^{2}} d A \\ R=\\{(x, y): 0 \leq x \leq \sqrt{3}, 1 \leq y \leq 2\\} \end{array} $$

Sketch the solid whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\) $$ \iint_{R}\left(25-x^{2}-y^{2}\right) d A $$

Use spherical coordinates to find the indicated quantity. Volume of the smaller wedge cut from the unit sphere by two planes that meet at a diameter at an angle of \(30^{\circ}\)

Find the Jacobian for the transformation from rectangular coordinates to spherical coordinates.