Exam 14: Iterated Integrals and Area in the Plane

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral. $\int_{-3}^{3} \int_{-\sqrt{9-x^{2}}}^{\sqrt {9-x^2}}\int_{x^2+y^2}^{9} x d z d y d x$

Set up a triple integral for the volume of the solid bounded by the coordinate planes and the plane given below. $z = 12 - 6 x - 8 y$

Evaluate the following iterated integral. $\int _ { 0 } ^ { \frac { \pi } { 9 } }$$\int_{0} ^ {11 cos\theta}$ r dr d $\theta$

Use cylindrical coordinates to find the volume of the solid bounded above by and below by $z = 15 x ^ { 2 } + 15 y ^ { 2 }$ .

Find the mass of the triangular lamina with vertices $( 0,0 ) , ( 20,40 ) \text {, and } ( 40,0 ) \text { for }$ the density $\rho=k$

$\text { Given } f ( x , y ) = e ^ { - \left( x ^ { 2 } + y ^ { 2 } \right) / 2 } , R : x ^ { 2 } + y ^ { 2 } \leq 729 , x \geq 0 \text {, use polar coordinates to set }$ up and evaluate the double integral $\int _ { R } \int f ( x , y ) d A$ .

Find the average value of $f ( x , y , z ) = x ^ { 3 } + y ^ { 2 } + z ^ { 4 } \text { over the region } Q \text {, where } Q \text { is a }$ cube in the first octant bounded by the coordinate planes, and the planes $x = 4 , y = 1$ , and $z = 2$ . The average value of a continuous function $f ( x , y , z )$ over a solid region $Q$ is $\frac{1}{V} \iiint_{Q} f(x, y, z) d V$ where $V$ is the volume of the solid region $Q$ .

Use a double integral to find the area of the region inside the circle $r = 17 \cos \theta \text { and }$ outside the cardioid $r = 1 + 15 \cos \theta \text {. Round your answer to two decimal places. }$

$\text { Find the average value of } f ( x , y , z ) = x + y + z \text { over the region } Q \text {, where } Q \text { is a }$ tetrahedron in the first octant with vertices $( 0,0,0 ) , ( 18,0,0 ) , ( 0,18,0 )$ and $( 0,0,18 )$ . The average value of a continuous function $f ( x , y , z )$ over a solid region $Q$ is $\frac{1}{V} \iiint_{Q} f(x, y, z) d V$ , where $V$ is the volume of the solid region $Q$ .

$\text { The area of a region } R \text { is given by the iterated integral } \int _ { 0 } ^ { 1 } \int _ { - \sqrt { 1 - y ^ { 2 } } } ^ { \sqrt { 1 - y ^ { 2 } } } 15 d x d y \text {. Switch the }$ order of integration and show that both orders yield the same area. What is this area?

Find the Jacobian $\frac { \partial ( x , y ) } { \partial ( u , v ) } \text { for the following change of variables: }$ $x = \frac { 1 } { 4 } ( 9 u - 4 v ) , y = \frac { 1 } { 4 } ( 3 u + 3 v )$

Determine the location of the horizontal axis $y _ { a }$ for figure (b) at which a vertical gate in a dam is to be hinged so that there is no moment causing rotation under the indicated loading (see figure (a)). The model for $y _ { a }$ is $y _ { a } = \bar { y } - \frac { I _ { y } } { h A }$ where $\bar { y }$ is the $y$ -coordinate of the centroid of the gate, $I _ { y }$ is the moment of inertia of the gate about the line $y = \bar { y } , h$ is the depth of the centroid below the surface, and $A$ is the area of the gate.

Use a triple integral to find the volume of the solid shown below. $z = 12 x y , 0 \leq x \leq 4,0 \leq y \leq 2$

Find $I _ { z } \text { for the indicated solid with density function } p = k x y z \text {. }$

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

Evaluate the double integral below. $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 5 + 5 \sin \theta } \theta r d r d \theta$

Find the area of the portion of the surface $f ( x , y ) = 2 + x ^ { 2 } - y ^ { 2 } \text { that lies above the }$ region $R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$ . Round your answer to two decimal places.

Write a double integral that represents the surface area of $f ( x , y ) = 10 y + 9 x ^ { 2 } \text { over }$ the region R: triangle with vertices $( 0,0 ) , ( 4,0 ) , ( 4,4 )$ . Use a computer algebra system to evaluate the double integral. Round your answer to two decimal places.

Use a change of variables to find the volume of the solid region lying below the surface $z = ( 2 y + 5 x ) ^ { 2 } \sqrt { 2 y - 7 x }$ and above the plane region $R$ : region bounded by the parallelogram with vertices $( 0,0 ) , ( - 2,5 ) , ( 4,26 ) , ( 6,21 )$ . Round your answer to two decimal places.

Convert the integral below from rectangular coordinates to both cylindrical and spherical coordinates, and evaluate the simpler iterated integral. $\int_{0}^{9} \int_{0}^{\sqrt{81-x^{2}}} \int_{0}^{\sqrt{81-x^{2}-y^{2}}} \sqrt{x^{2}+y^{2}+z^{2}} d z d y d x$