Exam 15: Multiple Integrals

Evaluate the double integral by first identifying it as the volume of a solid. $\iint _ { R } ( 15 - 2 x ) d A , R = \{ ( x , y ) \mid 3 \leq x \leq 7,4 \leq y \leq 5 \}$

Calculate the iterated integral. $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { y } 9 \cos \left( y ^ { 2 } \right) d x d y$

Find the moment of inertia with respect to a diameter of the base of a solid hemisphere of radius 3 with constant mass density function $\rho ( x , y , z ) = 5$

Find the area of the surface S where S is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies to the right of the xz-plane and inside the cylinder $x ^ { 2 } + z ^ { 2 } = 9$

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.

Find the moment of inertia about the y-axis for a cube of constant density 3 and side length $6$ if one vertex is located at the origin and three edges lie along the coordinate axes.

Express the integral as an iterated integral of the form $\int _ { a } ^ { b } \int _ { u ( x ) } ^ { v ( x ) } \int _ { c ( x , y ) } ^ { d ( x , y ) } f d z d y d x$ where E is the solid bounded by the surfaces $x ^ { 2 } = 1 - y , z = 3 \text {, and } z = y \text {. }$ $\iiint _ { E } f ( x , y , z ) d V$

Use spherical coordinates. Evaluate $\iiint _ { B } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 2 } d V$ , where $B$ is the ball with center the origin and radius $5$ .

Evaluate the triple integral. Round your answer to one decimal place. $\iiint _ { z } 5 x d V E = \left\{ ( x , y , z ) \mid 0 \leq y \leq 3,0 \leq x \leq \sqrt { 9 - y ^ { 2 } } , 0 \leq z \leq y \right\}$

Find the mass of the solid S bounded by the paraboloid $z = 6 x ^ { 2 } + 6 y ^ { 2 }$ and the plane $z = 5$ if S has constant density 3.

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.

The double integral $\iint _ { R } \left( 1 - y ^ { 2 } \right) d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq y , 0 \leq y \leq 1 \}$ , gives the volume of a solid. Describe the solid.

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 xy-plane and below the plane $z = x + 4$ .

Find the mass of a solid hemisphere of radius 5 if the mass density at any point on the solid is directly proportional to its distance from the base of the solid.

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 $y = 2 , x y = 4 ; x = \frac { u } { v } , y = v$ .

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

Use the Midpoint Rule with four squares of equal size to estimate the double integral. $\iint _ { R } \cos \left( x ^ { 4 } + y ^ { 4 } \right) d A , R = \{ ( x , y ) \mid 0 \leq x \leq 0.4,0 \leq y \leq 0.4 \}$

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

Sketch the solid whose volume is given by the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 2 - x } \int _ { 0 } ^ { 2 - x - y } f ( x , y , z ) d z d y d x$

Use spherical coordinates to find the moment of inertia of the solid homogeneous hemisphere of radius $5$ and density 1 about a diameter of its base.