Exam 14: Iterated Integrals and Area in the Plane

Find the mass and center of mass of the lamina bounded by the graphs of the equations given below for the given density. $\rho = k$ $y = \sqrt { A ^ { 2 } - x ^ { 2 } } , y = 0$

Use a double integral in polar coordinates to find the volume of the solid in the first octant bounded by the graphs of the equations given below. $z = x ^ { 3 } y , x ^ { 2 } + y ^ { 2 } = 4$

Find the area of the surface of the portion of the plane $z = 3 - 6 x - 2 y$ in the first octant.

Use a triple integral to find the volume of the solid shown below. $z = 36 - x ^ { 2 } - y ^ { 2 } , z \geq 0$

Use spherical coordinates to find the z coordinate of the center of mass of the solid lying between two concentric hemispheres of radii 6 and 7, and having uniform density k.

Evaluate the following improper integral. $\int _ { 10 } ^ { \infty } \int _ { 0 } ^ { \frac { 11 } { x } } y d y d x$

Sketch the image S in the uv-plane of the region R in the xy-plane using the given transformation. x=2u+3v y=2v

Determine the location of the horizontal axis $y _ { a } \text { for figure (b) at which a vertical }$ 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 $y _ { a } = \bar { y } - \frac { I _ { \bar { y } } } { h A }$ where $\bar { y }$ is the $y$ -coordinate of the centroid of the gate, $L _ { y }$ 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.

Find the Jacobian $\frac { \partial ( x , y ) } { \partial ( u , v ) } \text { for the following change of variables: }$ $x = \frac { u } { v } , y = 5 u + v$

Set up a triple integral that gives the moment of inertia about the $z \text {-axis of the solid }$ region $Q$ of density given below. \rho(x,y,z)= Q=\{-4\leqx\leq4,-4\leqy\leq4,0\leqz\leq1-y\}

Suppose the temperature in degrees Celsius on the surface of a metal plate is $T ( x , y ) = 60 - 4 x ^ { 2 } - y ^ { 2 }$ , where $x$ and $y$ are measured in centimeters. Estimate the average temperature if $x$ varies between 0 and 2 centimeters and $y$ varies between 0 and 7 centimeters.

Find the Jacobian for the change of variables given below. $x = - 4 u - 4 v ^ { 2 } , y = 4 u + 5 v$

Use polar coordinates to describe the region as shown in the figure below:

Use an iterated integral to find the area of the region bounded by $\sqrt { x } + \sqrt { y } = 5 , \quad x = 0 , \quad y = 0$

Set up and evaluate a double integral required to find the moment of inertia, I, about the given line, of the lamina bounded by the graphs of the following equations. Use a computer Algebra system to evaluate the double integral. $y = 0 , y = \sqrt { x } , x = 4$ , line: $x = 3$ $\rho = c x$

Find the area of the portion of the surface $f ( x , y ) = \sqrt { 12 - x ^ { 2 } - y ^ { 2 } } \text { that lies above }$ the region $R = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 144 \right\}$ Round your answer to two decimal places.

Combine the sum of the two iterated integrals into a single integral by converting to polar coordinates. Evaluate the resulting iterated integral. $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x } \sqrt { x ^ { 2 } + y ^ { 2 } } d y d x + \int _ { 2 } ^ { 2 \sqrt { 2 } } \int _ { 0 } ^ { \sqrt { 8 - x ^ { 2 } } } \sqrt { x ^ { 2 } + y ^ { 2 } } d y d x$

Find the center of mass of the lamina bounded by the graphs of the equations $y = 7 x ^ { 2 } , y = 0$ , and $x = 6$ for the density $\rho = k x y$