Exam 12: Multiple Integrals

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } 6 x y ^ { 2 } \sin z d z d y d x$ .

Use polar coordinates to combine the sum $\int _ { \sqrt { 2 } } ^ { 2 } \int _ { \sqrt { 4 - x ^ { 2 } } } ^ { x } x y d y d x + \int _ { 2 } ^ { 2 \sqrt { 2 } } \int _ { 0 } ^ { x } x y d y d x + \int _ { 2 \sqrt { 2 } } ^ { 4 } \int _ { 0 } ^ { \sqrt { 16 - x ^ { 2 } } } x y d y d x$ into one double integral. Then evaluate the integral.

Suppose the volume of a solid is given by $V = \int _ { 0 } ^ { 3 } \int _ { 0 } ^ { ( 3 - z ) / 2 } \int _ { 0 } ^ { 4 - x ^ { 2 } } d y d x d z$ .(a) Sketch the solid whose volume is given by V . (b) Evaluate the integral to find the volume of the solid.

Find the x-coordinate of the center of mass of the lamina that occupies the part of the disk $x ^ { 2 } + y ^ { 2 } \leq 1$ in the first quadrant and has density function p(x, y) = xy.

Find the area of the surface cut from the cone $z = 1 - \sqrt { x ^ { 2 } + y ^ { 2 } }$ by the cylinder $x ^ { 2 } + y ^ { 2 } = y$ .

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below.

Evaluate the iterated integral $\int _ { - 1 } ^ { 2 } \int _ { 0 } ^ { 1 - x } 4 - y d y d x$ .

Let $f ( x , y ) = x ^ { 2 }$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into two subrectangles by the line $x = \frac { 1 } { 2 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the center of Rij. Calculate the double Riemann sum of f.

Rewrite the integral $\int _ { - 1 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - y ^ { 2 } } } x d x d y$ in terms of polar coordinates, then evaluate the integral.

Evaluate $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { \sqrt { 4 - x ^ { 2 } } } \int _ { 0 } ^ { 4 - x ^ { 2 } - y ^ { 2 } } x ^ { 2 } d z d y d x$ by changing to cylindrical coordinates.

Find the area of the part of the cone $4 x ^ { 2 } + 4 y ^ { 2 } = z ^ { 2 }$ that is above the region in the first quadrant bounded by the line y = x and the parabola $y = x ^ { 2 }$ .

Find the mass of the solid that occupies the region bounded by $x ^ { 2 } + y ^ { 2 } = 1$ , z = 2, and z = 0 and has density function $\rho ( x , y , z ) = z$ .

Let $f ( x , y ) = x y$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be partitioned into four subrectangles by the lines $x = \frac { 1 } { 2 }$ and $y = \frac { 1 } { 2 }$ , and let $\left( x _ { i } ^ { * } , y _ { j } ^ { * } \right)$ be the upper right corner of Rij. Calculate the double Riemann sum of f.

Rewrite $\iint _ { R } f ( x , y ) d A$ as an iterated integral with x as the variable of integration in the outer integral, where R is the region shown below.

Compute the Riemann sum for the double integral $\iint _ { R } 4 d A$ where $R = [ 0,6 ] \times [ 0,2 ]$ for the given grid and choice of sample points.

Evaluate the iterated integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { x } \int _ { 0 } ^ { y ^ { 2 } } d x d y d z$ .

Evaluate $\iint _ { D } \cos \left( x ^ { 2 } + y ^ { 2 } \right) d A$ , where $D = \left\{ ( x , y ) \mid x ^ { 2 } + y ^ { 2 } \leq 1 \right\}$ , the unit disk.

Find the surface area of the surface z = xy inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .

Evaluate the triple integral $\iiint _ { E } x d V$ , where $E = \{ ( x , y , z ) \mid 0 \leq x \leq y , 0 \leq y \leq 1,0 \leq z \leq 1 \}$ .

Find the Jacobian of the transformation x = 3u + v, y = u - 2w, z = v + w.