math questions

Math 141 Calculus II Section 4065

Fall 2024

1. (10 points) The equation of the red curve is y2 = x3. Find the volume generated when area

bounded by (0,0), (4,8), (4,0) is revolved about the x-axis. (similar to 5.1 #7-12)

2. (10 points) Using the same figure in problem #1, find the volume generated when the area bounded by (0,0), (4,8), (4,0) is revolved about the y-axis.

3. (10 points) (similar to 5.1 #14-22). a. Compute the volume of the inverted cone using an integral. Note: Use the vertex

of the cone as the origin point (0,0).

b. Does your answer match with the known volume of a right cone, namely Vol = 1

3 𝜋𝜋 𝑟𝑟2 ℎ ? (yes or no)

4. (10 points) Calculate the length of the curve y = sin(𝑥𝑥) 𝑓𝑓𝑓𝑓𝑟𝑟 0 ≤ 𝑥𝑥 ≤ 𝜋𝜋 2

(similar to 5.2 #1-29)

a. Plot the curve for 0 <= x <= 𝜋𝜋

2 .

b. Plot the line segment between (0,0) and (𝜋𝜋

2 ,1) on the same graph.

c. Using the Pythagorean theorem, what is the length of the line segment?

d. Compute the length of the curve by setting up the arc length integral and then getting a numerical answer using Desmos.

e. Is your numerical answer from (d) greater than your numerical answer from (c)? (yes or no)

5. (10 points) Parabolic mirrors and reflectors have the shape of a paraboloid of

revolution. Similar to 5.2 #36-45. Find the area of the surface when the graph of y = 𝑥𝑥

2

2 for 0 <= x <= 2 is rotated

about the y-axis.

a. Plot the graph using Desmos.

b. Compute the surface area analytically when rotated about the y-axis.

c. Check your answer with an integration program. Take and post a

screenshot of your answer.

d. Does your analytic answer from (b) match your computer answer from (c)? (yes or no)

6. (10 points) (similar to 5.2 #15-25) A curve is defined parametrically by x(t) = 2cos(t) y(t) = sin(t)

a. Plot the curve between t = 0 and t = pi using Desmos. Enter ( 2cos(t), sin(t) )

b. Set up the integral to compute the arc length of the curve.

c. Use the Desmos integration program to get a numerical value for the perimeter. Include a screenshot.

7. (10 points) (similar to 5.3 #1-13, Week 2 Summary #6) A tank in the shape of a

paraboloid of revolution is completely filled with water. The tank is 8 ft high. We want to compute the amount of work needed to pump the water to a point 4 feet above the top of the tank.

Start with the work water pumping equation shown below. Note: The origin (0,0,0) of the coordinate system is at the base of the trough. Work = ∫ 𝜌𝜌𝜌𝜌 𝑑𝑑(𝑣𝑣𝑓𝑓𝑣𝑣) ℎ(𝑧𝑧)𝑏𝑏

𝑎𝑎 Also note: z is the vertical direction in this problem, not y. What are the following quantities? a. 𝜌𝜌𝜌𝜌 = b. 𝑎𝑎 = c. b = d. 𝑑𝑑(𝑣𝑣𝑓𝑓𝑣𝑣) = A(x,y) dz

What is A(x,y)?

e. ℎ(𝑧𝑧) = f. Using the above values, analytically compute the work from the

integral.

g. Check your answer with a Simpson’s rule program.

h. Do your answers from (f) and (g) match? (yes or no)

8. (10 points) Find the centroid of the figure bounded by f(x) = (-x2 -2x -1) and g(x) = (x – 5) (similar to 5.4 #11-22)

a. Make a graph of the curves.

b. Compute M using an integral (not using Desmos or any integration program)

c. Compute My (use Desmos)

d. Compute xc (the x centroid)

e. Compute Mx (use Desmos)

f. Compute yc (the y centroid)

g. Draw your centroid on a plot with the figure. Attach a screenshot.

h. Does your centroid look like the center of gravity of the figure? (yes or no)

9. (10 points) A waterproof rectangular box has sides L = 60 in = 5 ft, and a height

H = 6 in = 0.5 ft, and a width W = 6 in = 0.5 ft It is filled with water (ρg = 62.5 lbs/ft3).

a. Before you do any problem involving work, you have to define a reference point or origin. In principle, the choice of an origin is arbitrary. But some choices make the problem easier than others. Choose and clearly state your origin point for this problem.

b. What is the volume of the box?

c. If the box is filled with water, what is weight of the water?

d. What are the coordinates of the centroid of the hexagonal block relative to your origin choice?

e. How much work is done pumping the water to a point 3 feet higher than the top of the box? (similar to 5.4 #31-34) Use the method of centroids.

10. (10 points) A hemi-sphere of radius 4 feet is filled with water 𝜌𝜌𝜌𝜌 = 62.5 𝑣𝑣𝑙𝑙𝑙𝑙/𝑓𝑓𝑡𝑡3. How much work is done pumping the liquid to a point 2 ft above the top of the sphere?

Hint: Put the origin of your x-y coordinates at the center of the circle. The centroid of a hemisphere is 3

8 𝑟𝑟.