Exam 12: Multiple Integrals

Let $f ( x , y ) = x ^ { 2 } y$ , and let $R = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ . Let R be its own partition, and let $\left( x _ { 1 } ^ { * } , y _ { 1 } ^ { * } \right)$ be the center of R. Calculate the double Riemann sum of f.

Find the Jacobian of the transformation $x = s e ^ { t }$ , $y = s e ^ { - t }$ .

Find the volume of the region above the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and below the hemisphere $z - 9 = \sqrt { 9 - x ^ { 2 } - y ^ { 2 } }$ .

Evaluate the double integral $\iint _ { R } x y d A$ , where $R = \{ ( x , y ) \mid 0 \leq x \leq y , 0 \leq y \leq 2 \}$ .

Compute $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } x y ^ { 2 } e ^ { x y ^ { 3 } } d x d y$ ..

Evaluate the iterated integral $\int _ { 0 } ^ { 2 } \int _ { 0 } ^ { x ^ { 2 } } ( x + 3 \sqrt { y } ) d y d x$ .

Use the Midpoint Rule to estimate $\iint _ { R } \left( x ^ { 2 } + y ^ { 2 } \right) d A$ over $R = \{ ( x , y ) \mid 0 \leq x \leq 2,0 \leq y \leq 2 \}$ partitioned by the lines x = 1 and y = 1. Then estimate the average value of $f ( x , y ) = x ^ { 2 } + y ^ { 2 }$ over R.

Suppose X and Y are random variables. Find k such that the function $f ( x , y ) = \left\{ \begin{array} { c l } 0.2 e ^ { - ( 05 x + \not y ) } & \text { if } x \geq 0 , y \geq 0 \\0 & \text { otherwise }\end{array} \right.$ is a joint density function.

Let T be the transformation given by x = 2u + v, y = u + 2v.(a) A region S in the uv-plane is given below. Sketch the image R of S in the xy-plane. (b) Find the inverse transformation $T ^ { - 1 }$ .(c) Evaluate the double integral $\iint _ { R } ( - x + 2 y ) ^ { 2 } \cos ( 2 x - y ) d A$ .

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

Sketch the solid whose volume is given by the triple integral $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { \sqrt { 1 - x ^ { 2 } } } \int _ { 0 } ^ { \sqrt { 2 - x ^ { 2 } - y ^ { 2 } } } d z d y d x$ .

Evaluate $\iint _ { R } \sqrt { 9 - y ^ { 2 } } d A$ where $R = [ - 2,2 ] \times [ 0,3 ]$ by first identifying it as the volume of a solid.

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

Evaluate the integral $\iint _ { R } \sqrt { x ^ { 2 } + 9 y ^ { 2 } } d A$ , where R is the region enclosed by the ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$ .

Let E be the solid that lies below the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = a ^ { 2 }$ and above the cone $\phi = \beta$ , where $0 < \beta < \frac { \pi } { 2 }$ . Find the value of the triple integral $\iiint _ { E } z d V$ .

Find the surface area of the part of the cone $z = \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 1 / 2 }$ lying inside the cylinder $x ^ { 2 } - 2 x + y ^ { 2 } = 0$ .

Compute the Jacobian of the transformation T given by $x = \sqrt { 2 } u - \frac { 1 } { \sqrt { 2 } } v$ , $y = \frac { 1 } { \sqrt { 2 } } u + \sqrt { 2 } v$ . Compute the area of the image of $S = \{ ( u , v ) \mid 0 \leq u \leq 1,0 \leq v \leq 1 \}$ and compare it to the area of S.

Describe the region sketched below both as a type I and as a type II region.

Find the surface area for the part of the plane $5 z = 3 x - 4 y$ that lies inside the elliptic cylinder $x ^ { 2 } + 2 y ^ { 2 } = 2$ .

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