assignment 61

AMERICAN UNIVERSITY OF THE MIDDLE EAST

MA166 – Analytic Geometry and Calculus II, Fall 2020, Section: O2

Assignment 3, Due: Dec 8th

Instructions: There are 4 questions. Please answer clearly, show all your work step by step.

Student Name/ID: ___________________________________________________________ Student Name/ID: ___________________________________________________________ Q1 [25 pts]: Justify your answers clearly.

a) [8 pts] Evaluate the integral ∫ (𝑥−3)

(𝑥−4)(𝑥+5) 𝑑𝑥

b) [10 pts] First use a substitution and then partial fractions to evaluate the integral ∫ 1

𝑥√𝑥+1 𝑑𝑥

c) [7 pts] When we calculate the integral of a rational function by partial fractions, in which

case(s) we get an inverse tangent function in the end? Explain your answer.

Q2 [25 pts]: Justify your answers clearly.

a) [10 pts] Evaluate the integral ∫ 𝑒−4𝑥𝑑𝑥 1

−∞ or show that it is divergent.

b) [8 pts] Evaluate the integral ∫ 𝑑𝑥

1−𝑥

1

0 or show that it is divergent.

c) [7 pts] What is an improper integral of Type I? Give an example and explain why it is improper

of Type I.

Q3 [25 pts]: Justify your answers clearly.

a) [10 pts] Sketch the region enclosed by the curves 𝑦 = 𝑥, 𝑦 = 1/𝑥2, and the line 𝑥 = 2 and

find its area.

b) [8 pts] Find the area between the curves 𝑦 = 𝑥2 and 𝑦 = 3𝑥.

c) [7 pts] Does the integral ∫ (𝑥 − 𝑥2)𝑑𝑥 1

0 represent the area of a region? If so, make a sketch of

the region.

Q4 [25 pts]: Justify your answers clearly.

a) [10 pts] Find the volume of the solid generated by revolving the region bounded by 𝑦 = √𝑥,

𝑦 = 2, and 𝑥 = 0 about the 𝑥-axis.

b) [8 pts] Find the volume of the solid generated by revolving about the 𝑥-axis the region

bounded by 𝑦 = 4 − 𝑥2 and the 𝑥-axis.

c) [7 pts] A region between two curves (where 𝑥 ≥ 0) is rotated about the 𝑥-axis. The cross-

sections of the resulting object are as follows. Draw the region that is rotated and write the

integral which gives the volume.