Use either the disk method or the washer method to calculate the volume of the solid formed by revolving the given region about the given axis.

Region bounded by ? = ? ? + ?, ? = ?, and ? = ? about the ?-axis.

. Region bounded by ? = ? ? + ?, ? = ?, and ? = ? about the line ? = ?.

Find the volume of the figure whose base is the region bounded by y = x^2 + 4, y = x^2, the y-axis, and the vertical line x =2, and whose cross sections are squares parallel to the x-axis.

Find the volume of the figure created by rotating the region enclosed by the graphs of y = x^2 and y = x^3

a) around the x-axis

b) around the y-axis

Use the Divergence Theorem to evaluate

F · dS,

where

F(x, y, z) = z²xi +

+ cos z

j + (x²z + y²)k

and S is the top half of the sphere

x² + y² + z² = 4.

(Hint: Note that S is not a closed surface. First compute integrals over S₁ and S₂, where S₁ is the disk

x² + y² ≤ 4,

oriented downward, and S₂ = S₁ ∪ S.)

Find the point on the line

y = 3x + 4

that is closest to the origin.

(x, y) =

Jack is riding his bicycle from home to LeConte College with a velocity given by v(t)= -t^4 + 4-t^3 - 500t^2 +2000t in feet away from his house per minute and where t is in minutes. If it takes jack 20 minutes to get to LeConte College, find the following:

a.) Estimation of S (integral) upper 20, lower 0: v(t)dt using left-hand sums with n=2 subdivisions. Why is this a bad estimate?

b.)Estimation of S (integral) upper 20, lower 0, v(t)dt using right hand sums with n=5 subdivisions. What does this number represent an estimate of?

c.) Find the exact distance Jack traveled by integration (assume jack is 0 feet from his house at time = 0)(include units).

A strawberry farmer will receive $37 per bushel of strawberries during the first week of harvesting. Each week after that, the value will drop $0.80 per bushel. The farmer estimates that there are approximately 129 bushels of strawberries in the fields, and that the crop is increasing at a rate of four bushels per week. When should the farmer harvest the strawberries (in weeks) to maximize their value? (Assume that "during the first week of harvesting" here means week 1.)

weeks

How many bushels of strawberries will yield the maximum value?

bushels

What is the maximum value of the strawberries (in dollars)?

How do you find the domain of:

f(g)= 5x^2+4

f(g)= 3x; -2<x<6

f(g)= (1) / 3x-6

f(g)= (x+2) / x^2-1

f(g)= x^4 / x^2+x-6

f(g)= sqrt (x+1)

f(g)= sqrt (x^2+9)

17) Graph f (x ) = (x+3)e^−x. Include any possible local minimums or maximums, any inflection points, horizontal asymptotes, and vertical asymptotes.
18) Find the dimensions of a rectangle with area 6400 ft^2 whose perimeter is as small as possible.
19) Find f (x ) if f '( x) = sin x + 5 cos x where f(0) = 2
20) A particle is moving in a straight line with acceleration a(t) = e^t − 3t at time t. If s(0) = 0 and v (0) = 6 , find it's position at time t, s(t) .

Use the​ price-demand equation to determine whether demand is​ elastic, is​ inelastic, or has unit elasticity at the indicated values of p.

x=f(p)=1500-5p^2; p=10

The Health Fare Cereal Company makes three cereals using wheat, oats, and raisins. The portions and profit of each cereal are the following:

The company has 2320 pounds of wheat, 1380 pounds of oats, and 700 pounds of raisins available. How many pounds of each cereal should it produce to maximize profit?

7. The depth of water in a bay changes due to the tide. On one particular day, the depth can be modeled by ? = 6 cos (2π/11) + 7, where ? is hours since midnight, and ? is depth in metres.

a) Calculate the depth of the water at midnight.

b) At what rate is the depth changing at 4:00 a.m.?

c) At what time, between midnight and noon is the water level rising fastest?

Consider the vector field F(x, y) = <3 + 2xy, x2 − 3y 2> (b) Evaluate integral (subscript c) F · dr, where C is the curve (e^t sin t, e^t cost) for 0 ≤ t ≤ π.

A box with a square base and open top must have a volume of 32000 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

Find the following:
1. First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x.] Simplify your formula as much as possible.

2. Next, find the derivative, A'(x).

3. Now, calculate when the derivative equals zero, that is, when A'(x)=0.

4. We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x).

5. Evaluate A"(x) at the x-value you gave above.