Exam 15: Multiple Integrals

Find the volume of the solid bounded in the first octanat bounded by the cylinder $z = 9 - y ^ { 2 }$ and the planes $x = 1$ .

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 the equations $x = 3 \sqrt { y }$ $x = 0$ and $y = 1 ,$ and having the mass density $\rho ( x , y ) = x y$

Use a double integral to find the area of the region R where R is bounded by the circle $r = 6 \sin \theta$

Evaluate the double integral $\iint _ { R } x e ^ { y } d A$ , where $R$ is the triangular region with vertices $( 0,0 ) \text {, }$ $( 5,5 )$ and $( 0,6 )$ .

Find the Jacobian of the transformation. $x = \frac { u } { 2 u + 5 v } , y = \frac { v } { 4 u - 5 v }$

Find the center of mass of the lamina of the region shown if the density of the circular lamina is four times that of the rectangular lamina.

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

Sketch the solid whose volume is given by the integral $\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { \pi / 2 } \int _ { 0 } ^ { 2 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ Evaluate the integral.

Evaluate the integral $\iint _ { R } \frac { x ^ { 2 } } { x ^ { 2 } + y ^ { 2 } } d A$ , where R is the annular region bounded by the circles $x ^ { 2 } + y ^ { 2 } = 9$ and $x ^ { 2 } + y ^ { 2 } = 16$ by changing to polar coordinates.

Find the area of the surface. The part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies above the plane $z = 1$ .

Evaluate the double integral. $\iint _ { D } 2 y ^ { 2 } d A$ , $D$ is triangular region with vertices $( 0,1 ) , ( 1,2 ) \text { and } ( 4,1 )$ .

Find the area of the surface S where S is the part of the surface $x = y z$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 16$

Use cylindrical coordinates to evaluate $\iiint _ { E } \sqrt { x ^ { 2 } + y ^ { 2 } } d V$ where E is the region that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ and between the planes $z = - 6 \text { and } z = 5$ . Round the answer to two decimal places.

Calculate the iterated integral. $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } 8 y \sin x d z d y d x$

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 $z = 2$ .

Identify the surface with equation $r ^ { 2 } + z ^ { 2 } = 4$

Find the area of the surface. The part of the surface $z = 4 - x ^ { 2 } - y ^ { 2 }$ that lies above the xy-plane.

Calculate the double integral. $\iint _ { R } \left( 12 x ^ { 2 } y ^ { 3 } - 20 x ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 4 \}$

Find the center of mass of the system comprising masses m_k located at the points P_k in a coordinate plane. Assume that mass is measured in grams and distance is measured in centimeters. m₁ = 4, m₂ = 3, m₃ = 2 P₁(-3, -3), P₂(0, 3), P₃(-2, -1)

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 $z = 2$ and $z = 5$