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Calculus 2 - Exam 1 ( Semester C )

Exam Overview

This is the first exam for calculus 2. The material covered in this exam includes area between curves, moment and center of mass of a planar lamina, work moving water out of a container, volume of revolution, arc length and linear motion.

 

Exam Details

Tools

Time

2.5 hours

Calculators

no

Questions

6

Formula Sheet(s)

2 pages, 8.5x11 or A4

Total Points

100

Other Tools

ruler for drawing graphs

Instructions:
- Show all your work.
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).
- Give exact, simplified answers.

Question 1

[15 points] A stone is thrown vertically into the air off the edge of a bridge 160ft above a river. If the initial velocity of the rock is 48ft/sec (upward) and the acceleration is a constant -32ft/sec2 (downward), find the following. a. The velocity v(t) after t seconds. b. The position s(t) of the stone above the river. c. The velocity at which the stone hits the water.

answer

solution

Question 2

[10 points] Calculate the length of the arc \(y=2x^{3/2}\) from \(x=1/3\) to \(x=8/3\).

answer

solution

Question 3

plot for question 3

[10 points] Consider the region bounded by \(y=\ln(x)\) and the line with slope \(m=(3+\ln(5))/5\) and y-intercept at \(x=-3\) above the x-axis as shown in the plot. Set up and but do not evaluate one integral that calculates the shaded area.

answer

solution

Question 4

[20 points] Consider the region R bounded by \(y=x^3\) and \(y=\sqrt{x}\). a. Calculate the area of the region R. Sketch a plot and shade the region. b. Calculate the moments Mx and My. c. Determine the center of mass.

answer

solution

Question 5 [ exam link ]

[20 points] For the region R bounded by the graphs of \(y=2x\), the x-axis and \(x=1\), consider the solid generated by revolving the region about the line \(y=3\). Sketch a different plot of region R for each method, clearly showing the region by shading, the axes of revolution, the representative rectangle, R, r, p and h. a. Set up the integral to calculate the volume using the disc-washer method. b. Set up the integral to calculate the volume using the shell-cylinder method. c. Calculate the volume of the solid by evaluating one of the integrals you set up in parts a and b.

answer

solution

Question 6

q6 plot

[25 points] A swimming pool is 20 m long and 10 m wide, with a bottom that slopes uniformly from a depth of 1 m at one end to a depth of 2 m at the other end. Assuming the pool is full, how much work is required to pump the water to a level 1 m above the top of the pool?

answer

solution

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