Exam 12: Multiple Integrals

Find the volume bounded above by the surface $z = x ^ { 2 } - y ^ { 2 }$ , $x \geq 0$ , below by the xy-plane, and laterally by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Under the transformation x = u + v, y = v - 2u, the image of the circle $x ^ { 2 } + y ^ { 2 } \leq 1$ is an ellipse. What is the area of that ellipse?

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

Evaluate the iterated integral $\int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 1 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ .

Find the Jacobian of the transformation $x = u ^ { 2 }$ , $y = v ^ { 3 }$ , when $u = \frac { 1 } { 2 }$ and v = 1.

Evaluate $\int _ { - 2 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \sqrt { 9 - x ^ { 2 } - y ^ { 2 } } d y d x$

Give good estimates of lower and upper bounds for $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { 2 } x y \sqrt { x ^ { 2 } + y ^ { 2 } } d x d y$ .

Find the mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and has density function p(x, y) = xy.

Use the change of variables $x = a u$ , $y = b v$ , $z = c w$ to evaluate $\iiint _ { E } y d V$ , where E is the solid enclosed by the ellipsoid $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } + \frac { z ^ { 2 } } { c ^ { 2 } } = 1$ .

A phonograph turntable is made in the shape of a circular disk of radius 6 inches with density function $p ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Find the polar moment of inertia $I _ { 0 }$ of the disk.

Show that the average value of $f ( x , y ) = a x + b y$ over $R = \{ ( x , y ) \mid - 1 \leq x \leq 1 , - 1 \leq y \leq 1 \}$ is 0 for all values of a and b.

Compute the Riemann sum for the double integral $\iint _ { R } x + 2 y d A$ where $R = [ 0,6 ] \times [ 0,2 ]$ for the given grid and choice of sample points.

(a) Sketch the solid whose volume is given by the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 2 - y } \int _ { 0 } ^ { 4 - y ^ { 2 } } d x d z d y$ .(b) Rewrite the integral in part (a) as an equivalent iterated integral (or integrals) in the order dz, dx, dy.(c) Rewrite the integral in part (a) as an equivalent iterated integral (or integrals) in the order dy, dx, dz.

Use the change of variables u = 2x - y, v = x + y to evaluate $\iint _ { R } ( 6 x - 3 y ) d A$ where R is the region bounded by 2x - y = 1, 2x - y = 3, x + y = 1, and x + y = 2.

Evaluate $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { x z } x y z ^ { 2 } d y d x d z$ .

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 1 } \cos \left( y ^ { 2 } \right) d y d x$ .

Find the centroid of $D = \left\{ ( x , y ) \mid \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } \leq 1 , y \geq 0 \right\}$ , the region enclosed by the upper half of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ .

Evaluate the double integral $\iint _ { R } ( x + 2 y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq x \}$ .

The region D in $\mathbb { R } ^ { 2 }$ shown below is bounded by x = 1, $y = e ^ { x }$ , and $y = 1 - x ^ { 2 }$ . (a) Compute $\iint _ { R } x d A$ by finding $\int _ { 0 } ^ { 1 } \int _ { 1 - x ^ { 2 } } ^ { e ^ { x } } x d y d x$ .(b) Write down the integral or integrals needed to compute $\iint _ { R } x d A$ with the order of integration reversed.

Calculate the iterated integral $\int _ { 1 } ^ { 2 } \int _ { 0 } ^ { n / 2 } y \sin ( x y ) d y d x$ .