Calculus II Quiz #3 February 27, 2002 Name____________________ R.  Hammack Score ______

(1) Use the Fundamental Theorem of Calculus to evaluate the following integrals. Simpify your answer as much as possible.
(a)  = 4+8-1+1 = 12

(b)   = 3(5) - 3 = 12

(c)  = sin(π/2)-sin(π/4) =

(d)
 Because the integrand has an absolute value, we need to analyze it in some detail. Think of the graph of It is a parabola that opens "up", and its x intercepts are 0 and 1. A graph based on this information is sketched on the right. As you can see, is positive on the interval (-1,0) and negative on (0,1). Thus we break up our integral as follows:

1/3 +1/2 -1/3 + 1/2 = 1

(2)
Find the area of the region under the graph of   between   and   .

(3) Suppose    Find the interval(s) on which F increases, and the interval(s) on which F decreases.
To answer this question we need to look at the sign of the derivative of F.  By F.T.C. II, .
The denominator of this expression is always positive, so the sign of F '(x) is controlled by the numerator.
As you can see, the nunerator is positive when x > 3, and negative when x < 3.
Therefore F(x) increases on the interval (3,∞), and decreases on (-∞, 3),