Exam 13: Double and Triple Integrals

The functions $x = x ( u , v )$ and $y = y ( u , v )$ are given to determine transformations from the xy-coordinate system to a uv-coordinate system. Find the Jacobian of the transformation. $x=uv$$y = v$

Find the volume between the graph of the given function and the specified rectangle. $f ( x , y ) = x y ^ { 2 }$ and $R = \{ ( x , y ) : - 1 \leq x \leq 1 \text { and } - 1 \leq \mathrm { y } \leq 1 \}$

Evaluate the double integral $\iint _ { R } x \cos ( x y ) d A$ where $R = \left\{ ( x , y ) : 0 \leq x \leq \frac { \pi } { 2 } \text { and } 0 \leq \mathrm { y } \leq 1 \right\}$

Set up $\iint _ { R } f ( x , y ) d A$ as an iterated integral (or more, if necessary) where you integrate first with respect to $y$ , where $R = \{ ( x , y ) : 0 \leq x \leq 2 \text { and } 0 \leq \mathrm { y } \leq 2 x \}$

A disk of radius 2 meters is covered with mites. At the edge of the disk their density is 10,000 mites per square meter and at the center the density is 20,000 mites per square meter. If the mite density varies linearly with the distance from the center, how many mites are in the disk?

Give the cylindrical coordinates for the point with the spherical coordinates $\left( 4 , \frac { \pi } { 6 } , \frac { \pi } { 4 } \right)$

Find the volume of the solid bounded above by the plane $z = 10 - x - 2 y$ and below by the triangle with vertices (0, 0), (1, 0), and (0, 1) in the first quadrant of the xy plane.

Set up the double integral $\iint _ { R } ( x - y ) ^ { 2 } d A$ over the trapezoidal region with vertices (0, 2), (0, 3), (2, 0), and (3, 0) using the transformation u=x+y v=x-y

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Let the density at each point of T be equal to the point's distance from the x-axis. Find $M _ { x }$ for T.

Find the signed volume between the graph of the given function and the specified rectangle. $f ( x , y ) = x y + 1$ and $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { - 1 } ^ { 1 } \int _ { - \sqrt { 1 - x ^ { 2 } } } ^ { \sqrt { 1 - x ^ { 2 } } } f ( x , y ) d y d x$

Evaluate $\int _ { 3 } ^ { 17 } \int _ { - 2 - x } ^ { 2 + x ^ { 2 } } \int _ { 4 x } ^ { x ^ { 2 } } 3 x y z ^ { 3 } d z d y d x$

Rewrite the following integral, switching the order of the y and z integrations. $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { - \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } } f ( x , y , z ) d z d y d x$

Find the area enclosed by the spiral $r = \theta$ and the y-axis for $\frac { \pi } { 2 } \leq \theta \leq \frac { 3 \pi } { 2 }$

Evaluate the double integral $\iint _ { R } x ^ { 2 } y ^ { 3 } d A$ where $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$

The functions $x = x ( u , v )$ and $y = y ( u , v )$ are given to determine transformations from the xy-coordinate system to a uv-coordinate system. Find the Jacobian of the transformation. x=3u+2v y=u+v

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Let the density at each point of T be equal to 1. Find $I _ { x }$ for T.

Find the area between the cardioids $r = 5 + 5 \sin ( \theta )$ and $r = 2 + 2 \sin ( \theta )$

Let T be the triangle with vertices (0, 0), (2, 4), and (2, 0). Find the centroid of T.

Evaluate the following $\iiint _ { R } x ^ { 2 } y \sin ( z y ) d V$ , where $R = \left\{ ( x , y , z ) : 0 \leq x \leq 1,0 \leq y \leq 2 , \text { and } 0 \leq z \leq \frac { \pi } { 2 } \right\}$