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Calculus 2 - Exam A1

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17calculus > exam list > calc2 exam A1

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exam list

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

Exam Details

Tools

Time

1.5 hours

Calculators

see instructions

Questions

10

Formula Sheet(s)

none

Total Points

100

Other Tools

none

 

Downloads

download one page list of these questions

download multiple page exam with space to work out the solutions

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

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

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

answer

solution

Question 2

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

answer

solution

Question 3

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

answer

solution

Question 4

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

answer

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.

answer

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.

answer

solution

Section 3

Solve the following problems. Make sure your answers have correct units.

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.

answer

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.]

answer

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.
[ click here to show/hide the plot of G ]

Question 9

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

answer

solution

Question 10

(8 points) Set up, but do not evaluate, the integral for the area of the surface generated by revolving G about the x-axis.

answer

solution

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