Exam 15: Multiple Integrals

Use spherical coordinates to find the volume of the solid that lies within the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ above the xy-plane and below the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ . Round the answer to two decimal places.

Find the region E for which the triple integral $\iiint _ { E } \left( 1 - 5 x ^ { 2 } - 5 y ^ { 2 } - 5 z ^ { 2 } \right) d V$ is a maximum.

Find the volume bounded by the cylinders $x ^ { 2 } + y ^ { 2 } = 9$ and $y ^ { 2 } + z ^ { 2 } = 9$ .

Find the center of mass of a homogeneous solid bounded by the paraboloid $z = 100 - x ^ { 2 } - y ^ { 2 }$ and $z = 0$

Identify the surface with equation $\phi = \frac { \pi } { 8 }$

The joint density function for a pair of random variables $X$ and $Y$ is given. $f ( x , y ) = \left\{ \begin{array} { l l } C x ( 1 + y ) , & 0 \leq x \leq 2,0 \leq y \leq 2 \\0 , & \text { otherwise }\end{array} \right.$ Find the value of the constant $C$ .

Use polar coordinates to evaluate. $\int _ { - 4 } ^ { 4 } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } } } \sin \left( x ^ { 2 } + y ^ { 2 } \right) d x d y$

Find the area of the part of the plane $3 x + 3 y + z = 3$ that lies in the first octant.

Find an approximation for the integral. $\iint _ { R } \left( 4 x - 5 y ^ { 2 } \right) d A$ Use a double Riemann sum with $m = n = 2$ and the sample point in the upper right corner to approximate the double integral, where $R = \{ ( x , y ) \mid 0 \leq x \leq 8,0 \leq y \leq 4 \}$ .

Use the Midpoint Rule for double integrals with $m = n = 2$ to estimate the area of the surface. Round your answer to three decimal places. $z = x y + x ^ { 2 } + y ^ { 2 } , 0 \leq x \leq 5,0 \leq y \leq 5$

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $a = 15$ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. Assume the vertex opposite the hypotenuse is located at $( 0,0 )$ , and that the sides are along the positive axes.

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 rectangular region with vertices $( 0,0 ) \text {, }$ $( 5,0 )$ $( 5,4 )$ and $( 0,4 )$ , and having uniform density $\rho ( x , y ) = 9$

Determine whether to use polar coordinates or rectangular coordinates to evaluate the integral $\iint _ { R } f ( x , y ) d A$ , where f is a continuous function. Then write an expression for the (iterated) integral.

Use polar coordinates to find the volume of the solid under the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and above the disk $x ^ { 2 } + y ^ { 2 } \leq 9$ .

Describe the region whose area is given by the integral. $\int _ { 0 } ^ { \pi / 4 } \int _ { 0 } ^ { \cos 2 \theta } 2 r d r d \theta$

Evaluate the integral by changing to polar coordinates. $\iint _ { D } e ^ { - x ^ { 2 } - y ^ { 2 } } d A$ $D$ is the region bounded by the semicircle $x = \sqrt { 9 - y ^ { 2 } }$ and the $y$ -axis.

Find the mass and the center of mass of the lamina occupying the region R, where R is the region bounded by the graphs of $y = \sin 3 x$ $y = 0$ $x = 0$ and $x = \frac { \pi } { 3 }$ and having the mass density $\rho ( x , y ) = 4 y$

Find $\int _ { 0 } ^ { 5 } f ( x , y ) d x$ , if $f ( x , y ) = 2 x + 3 x ^ { 2 } y$ .

Find the mass of the lamina that occupies the region $D$ and has the given density function. Round your answer to two decimal places. $R = \{ ( x , y ) \mid 1 \leq x \leq 3,1 \leq y \leq 4 \} \rho ( x , y ) = 5 y ^ { 2 }$

Calculate the double integral. Round your answer to two decimal places. $\iint _ { R } \frac { 4 + x ^ { 2 } } { 1 + y ^ { 2 } } d A , R = \{ ( x , y ) \mid 0 \leq x \leq 6,0 \leq y \leq 4 \}$