Exam 12: Multiple Integrals

Evaluate the triple integral $\iiint _ { E } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) d V$ in spherical coordinates, where E is the solid in the first octant bounded by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and the coordinate planes.

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.

Evaluate the double integral $\iint _ { R } x d A$ , where $R = \left\{ ( x , y ) \mid y - 1 \leq x \leq \sqrt { 1 - y ^ { 2 } } , 0 \leq y \leq 1 \right\}$ .

Rewrite the integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { \sqrt { 3 } x } x d y d x$ in terms of polar coordinates, then evaluate the integral.

Evaluate the triple integral $\iiint _ { E } 1 d V$ in cylindrical coordinates, where $E = \{ ( r , \theta , z ) \mid 0 \leq r \leq 1,0 \leq \theta \leq \pi , 0 \leq z \leq 1 \}$ .

Use the change of variables x = 2u + 3v, y = 3u - 2v to evaluate $\iint _ { R } ( x + y ) d A$ , where R is the square with vertices (0, 0), (2, 3), (5, 1), and (3, -2).

Find the volume of the solid bounded by the paraboloid $z = 10 - 3 x ^ { 2 } - 3 y ^ { 2 }$ and the plane $z = 4$ .

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

A greenhouse is shown below. It is 10 ft wide and 20 ft long and has a at roof that is 12 ft high at one corner and 10 ft high at each of the adjacent corners. Find the volume of the greenhouse.

Evaluate the triple integral $\iiint _ { E } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) d V$ , where $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1 \right\}$ .

(a) Sketch the solid whose volume is given by the iterated integral $\int _ { - 2 } ^ { 2 } \int _ { x ^ { 2 } } ^ { 4 } \int _ { 0 } ^ { 4 - y } d z d y d x$ .(b) Rewrite the integral in part (a) as an equivalent iterated integral (or integrals) in the order dx, dz, dy.(c) Rewrite the integral in part (a) as an equivalent iterated integral (or integrals) in the order dy, dz, dx.

Let E be the solid under $z = 1 - y ^ { 2 }$ and above the region in the xy-plane bounded by x + y = 1 and x + y = 2. Sketch the solid, then express the volume of E as an iterated integral in rectangular coordinates.

Find $\iiint _ { S } x ^ { 2 } y d V$ , where S is the solid bounded by the cylinder $y = x ^ { 2 }$ and the planes z = 0, y = 1, and z = y.

Find the area of the part of the plane x +y + z = 6 that lies above the square with vertices (0, 0), (1, 0), (0, 1), and (1, 1).

Change the order of integration in the following integral and evaluate: $\int _ { 0 } ^ { 9 } \int _ { \sqrt { y } } ^ { 3 } \sin \left( \pi x ^ { 3 } \right) d x d y$ .

Give an example of a non-constant function f(x, y) such that the average value of f over $R = \{ ( x , y ) \mid - 1 \leq x \leq 1 , - 1 \leq y \leq 1 \}$ is 0.

Calculate the double Riemann sum of f for the partition of R given by the indicated lines and the given choice of $\left( x _ { i j } ^ { * } , y _ { i j } ^ { * } \right)$ . $f ( x , y ) = 2 x + x ^ { 2 } y$ , $R = \{ ( x , y ) \mid - 2 \leq x \leq 2 , - 1 \leq y \leq 1 \}$ , $x = - 1$ , $x = 0$ , $x = 1$ , $y = - \frac { 1 } { 2 }$ , $y = 0$ , $y = \frac { 1 } { 2 }$ ; $\left( x _ { i j } ^ { * } , y _ { i j } ^ { * } \right)$ = lower left corner of Rij.

Calculate the iterated integral $\int _ { 0 } ^ { \operatorname { In3 } } \int _ { 0 } ^ { \mathrm { In } 7 } e ^ { x - y } d x d y$ .

Find the Jacobian of the transformation $x = u \sin v$ , $y = u \cos v$ .

Let E be the solid under the paraboloid $z = 16 - x ^ { 2 } - y ^ { 2 }$ , above the xy-plane, and between the planes x = 2 and x = 3. Express the volume of E as an iterated integral in rectangular coordinates.