Exam 15: Multiple Integrals

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 8 y \sin x d z d y d x$

Use polar coordinates to find the volume of the solid bounded by the paraboloid $z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }$ and the plane $z = 1$ .

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

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 4$ .

Use cylindrical coordinates to evaluate $\iiint _ { z } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ where $E$ is the region that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ and between the planes $z = - 6$ and $z = 5$ . Round the answer to two decimal places.

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 $z = x + 4$ .

Estimate the volume of the solid that lies above the square $R = [ 0,4 ] \times [ 0,4 ]$ and below the elliptic paraboloid $f ( x , y ) = 68 - x ^ { 2 } - y ^ { 2 }$ . Divide $R$ into four equal squares and use the Midpoint rule.

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 2 \leq x \leq 5,3 \leq y \leq 8 \}$

Find the area of the surface $S$ where $S$ is the part of the surface $x = y z$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 25$ .

Use polar coordinates to evaluate $\int _ { - 3 } ^ { 3 } \int _ { 0 } ^ { \sqrt { 9 - x ^ { 2 } } } \sin \left( x ^ { 2 } + y ^ { 2 } \right) d x d y$

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

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

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 $y = 1$ .

Find the center of mass of a homogeneous solid bounded by the paraboloid $z = 25 - x ^ { 2 } - y ^ { 2 }$ and $z = 0$ .

Use the given transformation to evaluate the integral. $\iint _ { R } x y d A$ , where $R$ is the region in the first quadrant bounded by the lines $y = x , y = 3 x$ and the hyperbolas $x y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ .

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