# SOLVE PROBLEMS

Laykhalil

Math& 152 Hardy

Name: Midterm Exam I 10/30/2020

Instructions: You have until 11:59pm (Seattle time) on October 30th to complete the

following problems and upload your solutions to Canvas. You may print out and neatly write your solutions on this document or you may neatly write them out on a different document. Remember to show your work. Even if you are completely stumped by a problem, write down what you can. You are expected to do all integral and derivative calculations by hand. Blank answers will naturally be given no partial credit. Answers given without sufficient justification are considered incorrect. Your work will be graded on clarity and completeness, so make sure to justify your steps and carefully format your work in a way that I can understand!

1. Evaluate the following integrals.

𝐚. [4 pts] ∫ 𝑒𝑥(1 + 𝑒𝑥)𝑑𝑥

𝐛. [4 pts] ∫ sin(𝑡)

cos2(𝑡)

𝜋 3

0

𝑑𝑡

Math& 152 Hardy

Page 2 of 7

2. [10 pts] Evaluate the following integral, exactly, as the limit of a Riemann sum:

∫ (𝑥2 − 1)𝑑𝑥 6

0

Math& 152 Hardy

Page 3 of 7

3. [8 pts] Evaluate the following integral using geometry.

(Assume the curve is made up of circular arcs and straight line segments)

∫ 𝑓(𝑥) 5

−4

𝑑𝑥

Math& 152 Hardy

Page 4 of 7

4. [10 pts] Compute the total area enclosed by the curves 𝑦 = 𝑥3 − 14𝑥 − 10 and 𝑦 = −𝑥2 + 11𝑥 + 15.

(Reminder: Since this is a geometric question, treat all regions enclosed by the curves as having positive area)

Math& 152 Hardy

Page 5 of 7

5. [8 pts] Evaluate and simplify the following expression.

𝑑

𝑑𝑥 ∫ √𝑡3 + 1𝑑𝑡

cos(𝑥)

𝑒𝑥

Math& 152 Hardy

Page 6 of 7

6. [8 pts] The temperature, in degrees Fahrenheit, of a city t hours after 6 am is given by:

𝑇(𝑡) = 13 sin ( 𝜋𝑡

12 ) + 55

What is the average temperature between 9 am and 9 pm?

Math& 152 Hardy

Page 7 of 7

7. Let R be the region enclosed by 𝑦 = 2 ln(𝑥) , 𝑦 = 0, and 𝑥 = 4. Let S be the solid obtained by revolving R around the 𝑥-axis.

a. [2 pts] Sketch pictures of the region R and solid S. (This is not an art class! Don’t worry too much about perfection here.)

b. [3 pts] Set up and simplify, but do not evaluate, an integral describing the volume of S using the shell method.

c. [3 pts] Set up and simplify, but do not evaluate, an integral describing the volume of S using the washer method.