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Exam 13: Double and Triple Integrals

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Question 21

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Find the mass of the solid in the first octant bounded by the coordinate planes and the plane $x + 4 y + z = 4$ , where the density is equal to the distance from the xz-plane.

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Question 22

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Find the area enclosed in one petal of $r = \sin ( 3 \theta )$

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Question 23

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Find the area enclosed in one petal of $r = \sin ( 5 \theta )$

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

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 $M _ { y }$ for T.

Find the (signed) volume of the solid bounded by the given function over the specified region $\Omega$ . $f ( x , y ) = x y ^ { 2 }$ and $\Omega = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 4 \right\}$

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

Give the iterated integral as an iterated integral or sum of iterated integrals in the opposite order of integration. $\int _ { 0 } ^ { 1 } \int _ { - x } ^ { x } f ( x , y ) d y d x$

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

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

Evaluate the sum $\sum _ { i = 1 } ^ { 2 } \sum _ { j = 1 } ^ { 3 } i j$

Evaluate the integral $\int _ { 0 } ^ { 1 } \int _ { 2 x } ^ { x ^ { 2 } } x y d y d x$

Give the rectangular coordinates for the point with the spherical coordinates $\left( \sqrt { 3 } , \frac { \pi } { 4 } , \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) \right)$

Find the mass of the solid whose density is equal to twice the distance from the origin, which is outside the sphere of radius 3 and inside the sphere of radius 5.

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

Write the cylindrical equation $r = 16$ in rectangular coordinates.

Let D be the upper half of the unit disk. Assume it has density $\rho ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ Find the mass of D.

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

Set up the double integral $\iint _ { R } ( y - 2 x ) ^ { 3 } d A$ over the parallelogram with vertices (2, 1) , (3, 3) , (5, 2) , and (6, 4) using the transformation $\begin{array} { l } u = y - \frac { x } { 3 } \\v = y - 2 x\end{array}$

Give the cylindrical coordinates for the point with the rectangular coordinates $( 1,1,1 )$