Exam 15: Multiple Integrals

Sketch the solid bounded by the graphs of the equations $z = x ^ { 2 } + y ^ { 2 }$ and $z = 50 - x ^ { 2 } - y ^ { 2 }$ , and then use a triple integral to find the volume of the solid.

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

Use cylindrical coordinates to evaluate $\iiint _ { T } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ , where $T$ is the solid bounded by the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ and the planes $z = 1$ and $z = 5$ .

Find the area of the surface. The part of the surface $z = 36 - x ^ { 2 } - y ^ { 2 }$ that lies above the $x y$ -plane.

Use polar coordinates to find the volume of the solid inside the cylinder $x ^ { 2 } + y ^ { 2 } = 16$ and the ellipsoid $2 x ^ { 2 } + 2 y ^ { 2 } + z ^ { 2 } = 64$ .

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

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 $x = 4 u + 6 v , y = 6 u - 4 v$

Estimate the volume of the solid that lies above the square $R = [ 0,4 ] \times [ 0,4 ]$ and below the elliptic paraboloid $f ( x , y ) = 68 - x ^ { 2 } - y ^ { 2 }$ . Divide $R$ into four equal squares and use the Midpoint rule.

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 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 $z = x + 4$ .

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

Calculate the iterated integral. $\int _ { 1 } ^ { 4 } \int _ { 0 } ^ { 3 } ( 3 + 4 x y ) d x d y$

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 $D = \{ ( x , y ) \mid 0 \leq x \leq \pi , 0 \leq y \leq \sin ( x ) \}$ .

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

Evaluate the double integral by first identifying it as the volume of a solid. Select the correct answer. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 6,4 \leq y \leq 7 \}$

Find the mass of the lamina that occupies the region $D$ and has the given density function, if $D$ is bounded by the parabola $x = y ^ { 2 }$ and the line $y = x - 2$ . $\rho ( x , y ) = 5$ Select the correct answer.

Use the given transformation to evaluate the integral. $\iint _ { R } x y d A$ , where $R$ is the region in the first quadrant bounded by the lines $y = x , y = 3 x$ and the hyperbolas $x y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ .