Exam 16: Integrating Functions of Several Variables

Reverse the order of integration for the following integral. $\int _ { 0 } ^ { 2 } \int _ { y ^ { 2 } } ^ { \sqrt { 8 y } } f ( x , y ) c ^ { 2 } x d y$

Evaluate the integral $\int _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } d A$ , where R is the region shown below.

Evaluate the iterated integral $\int _ { \pi } ^ { 2 \pi } \int _ { 3 } ^ { 6 } r ^ { 4 } \sin \theta d r d \theta$

Find the volume under the graph of $f ( x , y ) = x e ^ { - 5 y }$ lying over the triangle with vertices (0, 0), (2, 2)and (4, 0).

Calculate the following integral: $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { y } x ^ { 5 } y ^ { 7 } d x y ^ { 2 } y$

Find the volume of the region under the graph of $f ( x , y ) = 1 + 2 x ^ { 2 } + 5 y$ and above the region $x ^ { 2 } \leq y , 0 \leq y \leq 4$

Evaluate the iterated integral. $\int_{1}^{2} \int_{0}^{\ln y} y e^{4 x} d x d y$

Evaluate $\int _ { W } z d V$ where W is the first octant portion of the ball of radius 3 centered at the origin.

The joint density function for x, y is given by $p ( x , y ) = \left\{ \begin{array} { l l } \frac { 5 } { 2 } x y ^ { 2 } & \text { if } 0 \leq x \leq 2 y \leq 2 \\0 & \text { otherwise }\end{array} \right.$ Find the probability that (x, y)satisfies $x \geq 1.1 y$

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e, $\pi$ , $\sqrt { 2 }$ , and so on.) $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { z } \int _ { 0 } ^ { y } x ^ { 2 } y ^ { 5 } z ^ { 5 } d x d y d z$

Evaluate the integral $\int _ { - 4 } ^ { 4 } \int _ { - \sqrt { 16 - y ^ { 2 } } } ^ { \sqrt { 16 - y ^ { 2 } } } \int _ { - \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } ^ { \sqrt { 16 - x ^ { 2 } - y ^ { 2 } } } \frac { 1 } { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d z c d x d y$ in spherical coordinates.

Consider the integral $\int _ { 0 } ^ { 3 } \int _ { - x } ^ { 0 } f ( x , y ) d y d x$ . (a)Sketch the region of integration and rewrite the integral with order of integration reversed. (b)Rewrite the integral in polar coordinates.

Let R be the region in the first quadrant bounded between the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the two axes.Then $\int _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A = \frac { \pi } { 8 }$ Let $\bar { R }$ be the region in the first quadrant bounded between the ellipse $25 x ^ { 2 } + 9 y ^ { 2 } = 1$ and the two axes. Use the change of variable x = s/5, y = t/3 to evaluate the integral $\int _ { x } \left( 75 x ^ { 2 } + 27 y ^ { 2 } \right) d A$

True or false? If f is any two-variable function, then $\int _ { R } f d A = 2 \int _ { S } f d A$ , where R is the rectangle 0 $\le$ x $\le$ 2, 0 $\le$ y $\le$ 1 and S is the square 0 $\le$ x, y $\le$ 1.

Find the triple integral of the function f(x, y, z)= xy sin (18yz)over the rectangular box 0 $\le$ x $\le$ $\pi$ , 0 $\le$ y $\le$ 1, 0 $\le$ z $\le$ $\pi$ /6.

Consider the region in 3-space bounded by the surface $f ( x , y ) = x ^ { 2 } + y ^ { 2 } - 1$ and the plane $z = k$ where $k > 0$ .Find the value of k such that the volume of this region below the xy-plane equals the volume of this region above the xy-plane.

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e, $\pi$ , $\sqrt { 2 }$ , and so on.) $\int _ { 3 } ^ { 4 } \int _ { 0 } ^ { y } y ^ { 2 } e ^ { x y } d x d y$

Suppose a solid is the region in three-space in the first octant bounded by the plane x + y = 1 and the cylinder $x ^ { 2 } + z ^ { 2 } = 1$ .If the density of this solid at a point (x, y, z)is given by $\delta ( x , y , z ) = z ^ { 3 }$ , find its mass.

A cylindrical tube of radius 2cm and length 3cm contains a gas.As the tube spins around its axis, the density of the gas increases as you get farther from the axis.The density, D, at a distance of r cm from the axis is D(r)= 1 +9 r gm/cc. Write a triple integral representing the total mass of the gas in the tube and evaluate the integral.

Let R be the region in the first quadrant bounded by the x- and y-axes and the line x + y = 7.Evaluate $\int _ { R } \sqrt { x + 2 y } d A$ exactly and then give an answer rounded to 4 decimal places.