Exam 15: Multiple Integrals

Use cylindrical coordinates to evaluate the triple integral $\iiint _ { { E } } x e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } }dV$ where E is the solid that lies between the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 8$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 18$ in the first octant.

A lamina occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 81$ in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Evaluate the integral by reversing the order of integration. $\int _ { 0 } ^ { 1 } \int _ { \arcsin y } ^ { \pi / 2 } 8 \cos x \sqrt { 1 + \cos ^ { 2 } x } d x d y$

Evaluate the integral by making an appropriate change of variables. Round your answer to two decimal places. $\iint _ { R } x y d A$ R is the parallelogram bounded by the lines $2 x - 3 y = - 5,2 x - 3 y = - 2,5 x + 2 y = - 5,5 x + 2 y = - 3$ .

Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ where B is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 8$

Evaluate the triple integral. Round your answer to one decimal place. $\iiint _ { F } 4 x y d V$ $E$ lies under the plane $z = 5 + x + y$ and above the region in the $x y$ -plane bounded by the curves $y = \sqrt { x } , y = 0$ , and $x = 4$ .

Find the area of the surface S where S is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ that lies inside the cylinder $x ^ { 2 } - x + y ^ { 2 } = 0$

Evaluate the iterated integral $\int _ { 0 } ^ { 4 } \int _ { \sqrt { x } } ^ { 2 } \cos y ^ { 3 } d y d x$ by reversing the order of integration.

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ .

Find the mass and the center of mass of the lamina occupying the region R, where R is the triangular region with vertices $( 0,0 ) \text {, }$ $( 2,5 )$ and $( 4,0 )$ , and having the mass density $\rho ( x , y ) = x$

Compute $\iint _ { D } \sqrt { 36 - x ^ { 2 } - y ^ { 2 } } d A$ , where $D$ is the disk $x ^ { 2 } + y ^ { 2 } \leq 36$ , by first identifying the integral as the volume of a solid.

Find the center of mass of the lamina that occupies the region D and has the given density function, if D is bounded by the parabola $y = 1 - x ^ { 2 }$ and the x-axis. $\rho ( x , y ) = 4 y$

Use spherical coordinate to find the volume above the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ and inside sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 a z$ .

Find the area of the surface. Round your answer to three decimal places. $z =$ $\frac { 4 } { 3 }$ $\left( x ^ { 2 / 3 } + y ^ { 2 / 3 } \right) , 0 \leq x \leq 5,0 \leq y \leq 3$

Calculate the iterated integral. $\int _ { 0 } ^ { \ln 4 } \int _ { 0 } ^ { \ln 5 } 3 e ^ { 5 x - y } d x d y$

Sketch the region of integration associated with the integral $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 8 \cos \theta } f ( r \cos \theta , r \sin \theta ) r d r d \theta$

Evaluate $\iiint _ { T } f ( x , y , z ) d V$ where $f ( x , y , z ) = 7 y$ and T is the region bounded by the paraboloid $y = x ^ { 2 } + z ^ { 2 }$ and the plane $y = 1$

Express the triple integral $\iiint _ { T } f ( x , y , z ) d V$ as an iterated integral in six different ways using different orders of integration where T is the solid bounded by the planes $x = 0$ $y = 0$ $z = 0$ and $x + 7 y + 8 z = 56$

Find the area of the surface S where S is the part of the plane $z = 2 x ^ { 2 } + y$ that lies above the triangular region with vertices $( 0,0 ) \text {, }$ $( 3,0 )$ , and $( 3,3 )$

Use polar coordinates to find the volume of the solid inside the cylinder $x ^ { 2 } + y ^ { 2 } = 16$ and the ellipsoid $6 x ^ { 2 } + 6 y ^ { 2 } + z ^ { 2 } = 64$ .