MATH 2B, Calculus
Sample Final
March, 2004
Instructor: Manuele Santoprete
Name:
Student I.D.:
Show your work: a correct answer without justifications may receive partial or no credit.
Non-programmable, non-graphic calculators are allowed.
Question 1
Evaluate the following indefinite integrals:
a. Zx
√x+ 1 dx b. Z√tan x
cos2xdx
Question 2
(a) If xsin πx =Rx2
0f(t)dt where fis a continuous function, find f(4). [Hint: Use the Fundamental
Theorem of Calculus].
(b) Compute the derivative of the following function
F(x) = Z√x
1/x
cos (t2)dt (x > 0).
Question 3
Sketch the region Denclosed by the curves y= sin x, y =x, x =π/2, x = 2. Set up the integral for
finding the area of the region D(decide whether to integrate with respect to xor to y). Find the area
of the region.
Question 4
Find the volume of the solid obtained by rotating the region bounded by y=x2, 0 ≤x≤2, y= 4,
x= 0 about the y-axis.
Question 5 Use logarithmic differentiation to compute dy
dx for the following function
3
v
u
u
u
tqtan−1(x) + cos(x2)
4
q3ln(x)−sin(x)
.
Question 6 Evaluate the following integrals using integration by parts:
(a)Zt3etdt (b)Zcos(ln(x))dx
1
