Chapter 13

The part of the saddle \(a z=x^{2}-y^{2}\) inside the cylinder \(x^{2}+y^{2}=a^{2}, a>0\)

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ S=\left\\{(x, y, z): 0 \leq x \leq \frac{1}{2} y, 0 \leq y \leq 4,0 \leq z \leq 2\right\\} $$

In Problems 13-18, 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. $$ \int_{0}^{2 \pi} \int_{1}^{3} r d r d \theta $$

In Problems \(7-14\), use cylindrical coordinates to find the indicated quantity. Center of mass of the homogeneous solid inside \(x^{2}+y^{2}=4\), outside \(x^{2}+y^{2}=1\), below \(z=12-x^{2}-y^{2}\), and above \(z=0\)

Evaluate the iterated integrals in Problems 1-14. \(\int_{\pi / 6}^{\pi / 2} \int_{0}^{\sin \theta} 6 r \cos \theta d r d \theta\)

In Problems \(11-16\), find the transformation from the uv-plane to the \(x y\)-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x^{2}-y^{2}, v=x+y $$

The partition of \(R\) into six equal squares by the lines \(x=2, x=4\), and \(y=2\). Approximate \(\iint_{R} f(x, y) d A\) by calculating the corresponding Riemann sum \(\sum_{k=1}^{6} f\left(\bar{x}_{k}, \bar{y}_{k}\right) \Delta A_{k}\), assuming that \(\left(\bar{x}_{k}, \bar{y}_{k}\right)\) are the centers of the six squares. \(f(x, y)=e^{x y}\)

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

In Problems 11-14, find the moments of inertia \(I_{x}, I_{y}\) and \(I_{z}\) for the lamina bounded by the given curves and with the indicated density \(\delta\). Triangle with vertices \((0,0),(0, a),(a, 0) ; \delta(x, y)=\) \(x^{2}+y^{2}\)

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ S=\left\\{(x, y, z): 0 \leq x \leq \sqrt{y}, 0 \leq y \leq 4,0 \leq z \leq \frac{3}{2} x\right\\} $$

In Problems 13-18, 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 15-22, use spherical coordinates to find the indicated quantity. Mass of the solid inside the sphere \(\rho=b\) and outside the sphere \(\rho=a(a

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

In Problems \(11-16\), find the transformation from the uv-plane to the \(x y\)-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x y, v=x $$

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

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

The part of \(z=9-x^{2}-y^{2}\) above the plane \(z=5\).

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{aligned} &S=\\{(x, y, z): 0 \leq x \leq 3 z \\ &\quad 0 \leq y \leq 4-x-2 z, 0 \leq z \leq 2\\} \end{aligned} $$

In Problems 13-18, 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. $$ \int_{0}^{\pi / 2} \int_{0}^{\cos \theta} r d r d \theta $$

In Problems 15-22, 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

In Problems 15-20, 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 \((1,4)\).

In Problems \(11-16\), find the transformation from the uv-plane to the \(x y\)-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x^{2}, v=x y $$

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

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

The part of \(z=9-x^{2}\) above the \(x y\)-plane with \(0 \leq x \leq 20\)

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ S=\left\\{(x, y, z): 0 \leq x \leq y^{2}, 0 \leq y \leq \sqrt{z}, 0 \leq z \leq 1\right\\} $$

In Problems 13-18, 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. $$ \int_{0}^{\pi} \int_{0}^{\sin \theta} r d r d \theta $$

In Problems 15-22, 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

In Problems 17-20, use a transformation to evaluate the given double integral over the region \(R\) which is the triangle with vertices \((1,0),(4,0)\), and \((4,3)\). $$ \iint_{R} \ln \frac{x+y}{x-y} d A $$

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

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

In Problems 11-20, 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 \((0,0,2)\).

In Problems 13-18, 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. $$ \int_{0}^{3 \pi / 2} \int_{0}^{\theta^{2}} r d r d \theta $$

In Problems 15-22, 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

In Problems 15-20, 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 \(y=3 x-x^{2} .\)

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

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

In Problems 11-20, 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.

In Problems 19-26, 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 \(x^{2}+y^{2}=4\)

In Problems 15-20, 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 \((0,2)\).

In Problems 17-20, use a transformation to evaluate the given double integral over the region \(R\) which is the triangle with vertices \((1,0),(4,0)\), and \((4,3)\). $$ \iint_{R} \sin (\pi(2 x-y)) \cos (\pi(y-2 x)) d A $$

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

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

Consider that part of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) between the planes \(z=h_{1}\) and \(z=h_{2}\), where \(0 \leq h_{1}

In Problems 11-20, 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 \(x=4\).

In Problems 19-26, 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 \(y=x\)

In Problems 15-22, 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

In Problems 17-20, 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 whose volume is given by the following double integrals over the rectangle \(R=\\{(x, y)\) : \(0 \leq x \leq 2,0 \leq y \leq 3\\}\). \(\iint_{R}\left(25-x^{2}-y^{2}\right) d A\)

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