Exam 15: Multiple Integrals

A swimming pool is circular with a $60$ -ft diameter. The depth is constant along east-west lines and increases linearly from $3$ ft at the south end to $9$ ft at the north end. Find the volume of water in the pool.

Find the area of the part of hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .

The sketch of the solid is given below. Given $a = 5$ , write the inequalities that describe it.

Use the transformation $x = \sqrt { 5 } u - \sqrt { \frac { 5 } { 3 } } v , y = \sqrt { 5 } u + \sqrt { \frac { 5 } { 3 } } v$ to evaluate the integral $\iint _ { R } \left( x ^ { 2 } - x y + y ^ { 2 } \right) d A$ , where R is the region bounded by the ellipse $x ^ { 2 } - x y + y ^ { 2 } = 5$ .

Find the volume of the given solid. Under the paraboloid $z = x ^ { 2 } + 3 y ^ { 2 }$ and above the rectangle $R = [ 0,3 ] \times [ 1,5 ]$ .

Find the mass and the moments of inertia $I _ { x }$ $I _ { y } ,$ and $I _ { 0 }$ and the radii of gyration $\bar {\bar { x} }$ and $\bar {\bar { y } }$ for the lamina occupying the region R, where R is the region bounded by the graphs of the equations $x = 2 \sqrt { y }$ $x = 0$ and $y = 2$ and having the mass density $\rho ( x , y ) = x y$

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 8 x y d x d y$

Express the volume of the wedge in the first octant that is cut from the cylinder $y ^ { 2 } + z ^ { 2 } = 4$ by the planes $y = x$ and $x = 7$ as an iterated integral with respect to $Z$ , then to $y$ , then to $x$ .

Identify the surface with equation $\rho \cos \phi = 10$

Use cylindrical coordinates to find the volume of the solid that the cylinder $r = 3 \cos \theta$ cuts out of the sphere of radius 3 centered at the origin.

Find the area of the part of the plane $2 x + 3 y + z = 9$ that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ .

Calculate the iterated integral. $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { 1 } 5 y e ^ { y y } d x d y$

Find the volume of the solid bounded by the surface $z = x \sqrt { 3 x ^ { 2 } + 3 y }$ and the planes $x = 1 , x = 0 , y = 1 , y = 0$ , and $z = 0$ . Round your answer to two decimal places.

Evaluate the double integral $\iint _ { R } ( 1 + 2 x + 8 y ) d A$ , where $R = \{ ( x , y ) \mid 0 \leq y \leq 1 , y \leq x \leq 3 y \} .$

Evaluate the integral $\int _ { 0 } ^ { 2 } \int _ { - \sqrt { 2 y - y ^ { 2 } } } ^ { \sqrt { 2 y - y ^ { 2 } } } y d x d y$ by changing to polar coordinates.

Use the given transformation to evaluate the integral. $\iint _ { R } ( x + y ) d A$ , where R is the square with vertices (0, 0), (4, 6), (6, $- 4$ ), (10, 2) and $x = 4 u + 6 v , y = 6 u - 4 v$

For which of the following regions would you use rectangular coordinates?

Find the mass of the solid E, if E is the cube given by $0 \leq x \leq 3,0 \leq y \leq 3,0 \leq z \leq 3$ and the density function $\rho$ is $\rho ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 }$ .

An agricultural sprinkler distributes water in a circular pattern of radius $100$ ft. It supplies water to a depth of $e ^ { - \gamma }$ feet per hour at a distance of $r$ feet from the sprinkler. What is the total amount of water supplied per hour to the region inside the circle of radius $15$ feet centered at the sprinkler?

The joint density function for random variables $X , Y$ and $Z$ is $f ( x , y , z ) = C x y z$ for $0 \leq x \leq 2,0 \leq y \leq 4,0 \leq z \leq 1$ and $f ( x , y , z ) = 0$ otherwise. Find the value of the constant $C$ . Round the answer to the nearest thousandth.