Exam 15: Multiple Integrals

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 12 x y d x d y$

Find the volume of the solid bounded in the first octanat bounded by the cylinder $z = 9 - y ^ { 2 }$ and the planes $x = 1$ .

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius 5 .

Find the mass of the lamina that occupies the region $D$ and has the given density function. Round your answer to two decimal places. $R = \{ ( x , y ) \mid 1 \leq x \leq 3,1 \leq y \leq 4 \} \rho ( x , y ) = 5 y ^ { 2 }$

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

Evaluate the iterated integral by converting to polar coordinates.Round the answer to two decimal places. $\int _ { - 5 } ^ { 5 } \int _ { 0 } ^ { \sqrt { 25 - y ^ { 2 } } } \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 / 2 } d x d y .$

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { y y } d x d y$

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral $\iint _ { R } f ( x , y ) d A$ , where $f$ is a continuous function. Then write an expression for the (iterated) integral.

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 4 y \sin x d z d y d x$

Find the area of the surface $S$ where $S$ is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies inside the cylinder $x ^ { 2 } - 4 x + y ^ { 2 } = 0$ .

Evaluate the integral $\iint _ { R } \frac { x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } d A$ , where $R$ is the annular region bounded by the circles $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 16$ , by changing to polar coordinates.

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ , where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 1$ and $z = 5$ .

Describe the region whose area is given by the integral. $\int _ { 0 } ^ { \pi / 4 } \int _ { 0 } ^ { \cos 2 \theta } 2 r d r d \theta$

Use spherical coordinates to evaluate $\iiint _ { B } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ , where $B$ is the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 10$ .

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

Evaluate the integral $\iiint _ { B } f ( x , y , z ) d V$ where $f ( x , y , z ) = x y ^ { 2 } + y z ^ { 2 }$ and $B = \{ ( x , y , z ) \mid 0 \leq x \leq 2 , - 1 \leq y \leq 1,0 \leq z \leq 3 \}$ with respect to $x , y$ , and $z$ , in that order.

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the region bounded by the graphs of $y = \sin 3 x , y = 0 , x = 0$ , and $x = \frac { \pi } { 3 }$ , and having the mass density $\rho ( x , y ) = 4 y$

Use cylindrical coordinates to evaluate the triple integral $\iiint _ { E } y d V$ where $E$ is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the $x y$ -plane and below the plane $z = x + 4$ .