Let W be the region bounded by the planes $x = 0 , y = 0 , z = 0 , x + y = 1$ , and $z = x + y$ (a) Find the volume of W. (b) Evalute $\displaystyle \iiint _ { W } x d x d y d z$ (c) Evalute $\displaystyle \iiint _ { W } y d x d y d z$

Let R be the region bounded by the following curves. Let S be the solid generated when R is revolved about the given axis. If possible, find the volume of S by both the disk/washer and shell methods. Check that your results agree and state which method is easier to apply. y=

$$( x - 2 ) ^ { 3 }$$

-2, x=0, and y=25; revolved about the y-axis

At time 0, a coin that comes up heads with probability p is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate

$$\lambda$$

, the coin is picked up and flipped. (Between these times the coin remains on the ground.) What is the probability that the coin is on its head side at time t? Hint What would be the conditional probability if there were no additional flips by time t, and what would it be if there were additional flips by time t?

Find the volume of the solid bounded by the planes x+y=1, x-y=1, x=0, z=0, and z=10.

The local movie theater charges $8 for adults and$5.50 for children. How much is the cost for 82 adults and 153 children to watch a movie?

sketch the graph of the function. (Include two full periods.) y = 1/2 sec πx

$$A=\left[\begin{array}{lll}{2} & {0} & {0} \\ {0} & {4} & {0} \\ {0} & {0} & {5}\end{array}\right]$$

. Find $A^{-1}$.

Describe the given region as an elementary region.The region between the cone . $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and the paraboloid $z = x ^ { 2 } + y ^ { 2 }$

Suppose that 10 balls are put into 5 boxes, with each ball independently being put in box i with probability

$$p _ { i } , \sum _ { i = 1 } ^ { 5 } p _ { i } = 1$$

. (a) Find the expected number of boxes that do not have any balls. (b) Find the expected number of boxes that have exactly 1 ball.

Evaluate the following triple integral: $\displaystyle \iiint _ { W } \sin x \ d x \ d y \ d z$ where W is the solid given by $0 \leq x \leq \pi , 0 \leq y \leq 1$ and $0 \leq z \leq x$

Evaluate the given functions. f(T) = 7.2 - 2.5|T|; find f(2.6) and f(-4)

Find the volume of the region determined by $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 10 , z \geq 2$ Use the disk method from one-variable calculus and state how the method is related to Cavalieri’s principle.

Let $f$ and $g$ be functions from $\mathbb{R}^3$ to $\mathbb{R}$. Suppose $f$ is differentiable and $\nabla f (\mathbf{x}) = g(\mathbf{x})\mathbf{x}$. Show that spheres centered at the origin are contained in the level sets for $f$ ; that is, $f$ is constant on such spheres.

The number of celebrants at the Independence Day party had increased by 25% over last year. If there were 2220 people gathered this year, how many were there last year?