Exam 14: Iterated Integrals and Area in the Plane

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equations given below. $z = \frac { 1 } { 25 + y ^ { 2 } } , x = 0 , x = 2 , y \geq 0$

Use a change of variables to find the volume of the solid region lying below the surface $z = \sqrt { ( x - y ) ( x + 3 y ) }$ and above the plane region $R$ : region bounded by the parallelogram with vertice $( 0,0 ) , ( 5,5 ) , ( 20,0 ) , ( 15 , - 5 )$ . Round your answer to two decimal places.

Find a such that the volume inside the hemisphere $z = \sqrt { 100 - x ^ { 2 } - y ^ { 2 } } \text { and outside }$ the cylinder $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ is one-half the volume of the hemisphere. Round your answer to four decimal places.

Set up a double integral that gives the area of the surface on the graph of $f ( x , y ) = 10 \cos \left( x ^ { 2 } + y ^ { 2 } \right)$ over the region $R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq \frac { \pi } { 4 } \right\}$

Use a double integral to find the area enclosed by the graph of $r = 3 \cos 3 \theta$

The area of a region R is given by the iterated integrals $\int _ { 0 } ^ { 2 x } 16 d y d x + \int _ { 2 } ^ { 4 } \int _ { 0 } ^ { 4 - x } 16 d y d x$ Switch the order of integration and show that both orders yield the same area. What is this area?

Use a double integral to find the volume of the indicated solid.

Set up a double integral that gives the area of the surface of the graph of f over the region R. f(x,y)=-9xy+9 R=\{(x,y):-6\leqx\leq6,-8\leqy\leq8\}

Use spherical coordinates to find the z coordinate of the center of mass of the solid lying between two concentric hemispheres of radii 4 and 7, and having uniform density k.

Use cylindrical coordinates to find the volume of the solid inside the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64$ and above the upper nappe of the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 }$ .

Sketch the solid whose volume is given by the iterated integral given below and use the sketch to rewrite the integral using the indicated order of integration. $\int_{0}^{2} \int_{y}^{2}\int_{0}^{\sqrt{4-y^2}} dzdxdy$ Rewrite the integral using the order $d z d y d x$ .

Find the center of mass of the rectangular lamina with vertices $( 0,0 ) , ( 14,0 ) , ( 0,12 )$ , and $( 14,12 )$ for the density $\rho = k \left( x ^ { 2 } + y ^ { 2 } \right)$

Find $\bar z$ of the center of mass of the solid of given density $\rho ( x , y , z ) = k x \text { bounded }$ by the graphs of the equations $Q : z = 12 - x , z = 0 , y = 0 , y = 4 , x = 0$ .

Find the area of the surface for the portion of the paraboloid $z = 64 - x ^ { 2 } - y ^ { 2 }$ in the first Octant.

Use an iterated integral to find the area of the region bounded by $2 x - 5 y = 0 , \quad x + y = 6 , \quad y = 0 .$

A company produces a spherical object of radius 24 centimeters. A hole of radius 5 centimeters is drilled through the center of the object. Find the outer surface area of the object.

Find the mass of the lamina bounded by the graphs of the equations $y = \sqrt { 441 - x ^ { 2 } }$ and $0 \leq y \leq x$ for the density $\rho = k$

Evaluate the following integral. $\int _ { y } ^ { \frac { \pi } { 5 } } \sin ^ { 3 } x \cos y d x$

Find the mass of the triangular lamina with vertices $( 0,0 ) , ( 18,36 ) \text {, and } ( 36,0 ) \text { for }$ the density $\rho=k x y$

Find the area of the surface given by $z = f ( x , y ) \text { over the region } R \text {. }$ $f ( x , y ) = 3 - 2 x + 5 y$ $R$ : square with vertices $( 0,0 ) , ( 4,0 ) , ( 4,4 ) , ( 0,4 )$