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Math 101: Fall 2011 Exam 1, Exams of Calculus

The fall 2011 exam for math 101, including problems related to indefinite and definite integrals, sketching regions, setting up integrals for volumes, and solving an initial value problem. Students are required to show their work for full credit.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

parni
parni 🇮🇳

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Name:
Math 106: Fall 2011
Exam 1: September 30
Please write your final answer in the space provided. For full credit you must show your work.
Good Luck!
1. Evaluate the indefinite integrals.
(a) (5 points) Z1
xdx (1a)
(b) (8 points) Zcos x
sin4xdx (1b)
2. (8 points) Evaluate the definite integral exactly (i.e., there should be no decimals in your answer).
Z7
6
3x3
13 + x4dx
(2)
1
pf3
pf4
pf5

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Name:

Math 106: Fall 2011

Exam 1: September 30

Please write your final answer in the space provided. For full credit you must show your work. Good Luck!

  1. Evaluate the indefinite integrals.

(a) (5 points)

x dx (1a)

(b) (8 points)

∫ cos x sin^4 x

dx (1b)

  1. (8 points) Evaluate the definite integral exactly (i.e., there should be no decimals in your answer). ∫ √ 7 √ 6

3 x^3 √ 13 + x^4

dx

(2)

  1. (5 points) Name the property of the pictured function f (x) that guarantees that M 373 ≤

∫ (^3)

− 2

f (x) dx.

  1. (8 points) The following table shows the measured rate of water flow f (t) (in liters per minute) out of a tank during a 10 minute interval. t 0 1 2 3 4 5 6 7 8 9 10 f (t) 250 300 325 330 325 290 250 200 160 110 80

The total amount of water that flows out of the tank is given by I =

∫ (^10)

0

f (t) dt. Estimate I using M 5. (4)

  1. (8 points) Write the definite integral(s) that equals the length of the curve y = xex^ from x = −2 to x = 3. (Set up the integral only. Do NOT evaluate it.)
  1. Let R be the region bounded by y = x^2 and y = 2x.

(a) (8 points) Set up the integral you would use to find the volume of the solid generated by revolving R around the x-axis.

(7a)

(b) (8 points) Set up the integral you would use to find the volume of the solid generated by revolving R around the line x = 3.

(7b)

  1. (10 points) Consider the Initial Value Problem:

dy dt = y^2 t, y(1) = 2. Use Euler’s Method with 3 steps to estimate y(1.75). Do this by hand (writing out all steps and using a calculator only to perform basic operations). Round to 4 digits after the decimal point.