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## Calculus 2 - Exam A1

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This is the first exam for second semester single variable calculus.

 Downloads Exam Details Tools Time 1.5 hours Calculators see instructions Questions 10 Formula Sheet(s) none Total Points 100 Other Tools none download one page list of these questions

Instructions:
- This exam is in two main parts, labeled parts A and B, with different instructions for each part.
- 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).

#### Part A - Questions 1-4

Instructions for Part A - - You have 30 minutes to complete this part of the exam. No calculators are allowed. Show all your work, justify your conclusions and give exact, completely factored answers.

Section 1

Evaluate the following integrals. Each question in this section is worth 9 points.

Question 1

$$\displaystyle{ \int_{0}^{1}{x^2e^{-4x}~dx} }$$

solution

Question 2

$$\int{\sin^3(\pi t) \cos^3 (\pi t)~dt}$$

solution

Question 3

$$\displaystyle{ \int_{0}^{2}{\frac{3}{(r^2+4)^{3/2}}dr} }$$

solution

Question 4

$$\displaystyle{ \int{\frac{9x^2+6x+12}{(x+1)(x^2+4)} dx} }$$

solution

#### Part B - Questions 5-10

Instructions for Part B - - You may use your calculator. You have one hour to complete this part of the exam. Show all your work. Correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions. Give exact answers.

Section 2 [16 points]

A region $$R$$ is bounded by the graph of $$y = e^x$$, the line $$y=1$$ and the line $$x=3$$.
Sketch the region $$R$$. [ click here to show/hide the plot ]

Question 5

A solid is generated by revolving the region $$R$$ about the line $$x=-1$$. Set up, but do not evaluate, a definite integral for the volume of the resulting solid of revolution.

solution

Question 6

A second solid has base $$R$$ and cross-sections perpendicular to the x-axis are disks with a diameter in $$R$$. Set up, but do not evaluate, a definite integral for the volume of this region.

solution

Section 3

Question 7

(16 points) A thin metal plate in the plane occupies the triangle with vertices $$(0,2)$$, $$(0,0)$$ and $$(2,0)$$. Find the centroid of the plate. You may use symmetry and known area or volume formulas from geometry to evaluate integrals.

solution

Question 8

(16 points) An upright water tank has a flat base in the plane and is filled with water. The tank has height $$7$$ feet, and, for $$0 \le y \le 7$$, the cross-section parallel to the base at height $$y$$ has area $$y^2+2$$. How much work is done in pumping all the water over the top edge of the tank? [Use $$\delta~ lb/ft^3$$ for the weight-density of water.]

solution

Section 4

Let G be the portion of the graph with parametric equations
$$x= 1+\sin(t)$$     $$y = 3+2\cos(t)$$    for $$0 \le t \le 2\pi$$.
Plot G and answer the following questions based on the graph.

Question 9

(8 points) Set up, but do not evaluate, the integral for the length of G.