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.