Exam 15: Multiple Integrals

Find the mass and the center of mass of the lamina occupying the region $R$ , where $R$ is the triangular region with vertices $( 0,0 ) , ( 5,2 )$ , and $( 10,0 )$ , and having the mass density $\rho ( x , y ) = x$ .

The sketch of the solid is given below. Given $a = 7$ , write the inequalities that describe it.

Use the transformation $x = \sqrt { 2 } u - \sqrt { \frac { 2 } { 3 } } v , y = \sqrt { 2 } u + \sqrt { \frac { 2 } { 3 } } v$ 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 $x ^ { 2 } - x y + y ^ { 2 } = 2$ .

Use a computer algebra system to find the moment of inertia $I _ { 0 }$ of the lamina that occupies the region $D$ and has the density function $\rho ( x , y ) = 3 x y$ , if $D = \{ ( x , y ) \mid 0 \leq x \leq \pi , 0 \leq y \leq \sin ( x ) \}$ .

Evaluate the iterated integral. $\int _ { 1 } ^ { 3 } \int _ { y } ^ { 3 } 10 x y d x d y$

An electric charge is spread over a rectangular region $R = \{ ( x , y ) \mid 0 \leq x \leq 3,0 \leq y \leq 2 \}$ . Find the total charge on $R$ if the charge density at a point $( x , y )$ in $R$ (measured in coulombs per square meter) is $\sigma ( x , y ) = 4 x ^ { 2 } + y ^ { 3 }$ .

Use cylindrical coordinates to evaluate the triple integral $\iiint _ { E } y d V$ where $E$ is the solid that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 3$ and $x ^ { 2 } + y ^ { 2 } = 7$ above the $x y$ -plane and below the plane $z = x + 4$ .

Use polar coordinates to find the volume of the solid bounded by the paraboloid $z = 7 - 6 x ^ { 2 } - 6 y ^ { 2 }$ and the plane $z = 1$ .

Find the center of mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $y = 1 - x ^ { 2 }$ and the $x$ -axis. $\rho ( x , y ) = 4 y$

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ .

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 $x ^ { 2 } + y ^ { 2 } = 36$ .

Find the center of mass of a homogeneous solid bounded by the paraboloid $z = 25 - x ^ { 2 } - y ^ { 2 }$ and $z = 0$ .

Use a triple integral to find the volume of the solid bounded by $x = y ^ { 2 }$ and the planes $z = 0$ and $x + z = 3$ .

A swimming pool is circular with a $20 - \mathrm { ft }$ diameter. The depth is constant along east-west lines and increases linearly from $3 \mathrm { ft }$ at the south end to $9 \mathrm { ft }$ at the north end. Find the volume of water in the pool.

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 25$ .

Find the mass and the moments of inertia $I _ { x } , I _ { y }$ , and $I _ { 0 }$ and the radii of gyration $\overline{\bar { x }}$ and $\overline {\bar { y }}$ 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 $\rho ( x , y ) = x y$ .

Calculate the iterated integral. $\int _ { 0 } ^ { \ln 4 } \int _ { 0 } ^ { \ln 5 } 3 e ^ { 5 x - y } d x d y$

Use cylindrical or spherical coordinates, whichever seems more appropriate, to evaluate $\iiint _ { E } z d V$ where $E$ lies above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the plane $z = 16$ .

Find the volume of the solid bounded by the surface $z = 5 + ( x - 4 ) ^ { 2 } + 2 y$ and the planes $x = 3 , y = 4$ and coordinate planes.