Homework 4. Calculus D

Due Friday August 1st

Exercise 1

Find antiderivatives for the functions
f(x) = (2 x^3 + 4 x^2 + 2 x + 1)/(x^4 + x^2)
g(x) = (5 x^2 - 2 x + 1)/(x^3 - x^2 + x - 1)

Exercise 2

Let f(x) = sin(x) cos(x).
a) Compute the indefinite integral of f by substitution.
b) Now compute the same integral as in a) by noting that sin(2x) = 2 sin(x) cos(x)
c) using either a) or b) compute the definite integral of f between 0 and \pi = 3.1415....
d) Apparently, the answers of a) and b) seem to be different, can you explain this?

Exercise 3

Compute the area bounded by the curves y = x^n and y = x^{1/n} (that is the n-th root of x). Check the signs in your computations and see that in the interval that we are interested, x^n is smaller than x^{1/n}. [Of course, n is a natural number here and bigger than 1, the answer should depend on n].

Exercise 4

Solve problem 44 in Edwards and Penney section 6.4 (this is not trivial)

Exercise 5

Compute the integrals 2) 14) 28) and 34) in section 9.3 (pp. 524).

Exercise 6

a) Find the length of the arc of the parabola y^2 = x from (0,0) to (1,1)
b) Find the area of the surface that results from rotating the curve in a) around the x axis.

Reimundo Heluani

Last modified: Sun Jul 25 11:55:43 EDT 2004