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Calculus  II                                                   Quiz #3                                                  February 26, 2003

Name____________________               R.  Hammack                                                   Score ______
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(1) Use the Fundamental Theorem of Calculus to evaluate the following integrals. Simpify your answer as much as possible.

(a)  ∫_ (-3)^0 (x^2 - 4x + 7) dx =[x^3/3 - 2x^2 + 7x] _ (-3)^0 = (0^3/3 - 2 (0^2) + 7 (0)) - ((-3)^3/3 - 2 (-3)^2 + 7 (-3)) = 48


(b)   ∫_0^(π/4) sec^2(x) dx =[tan(x)] _0^(π/4) = tan(π/4) - tan(0) = 1 - 0 = 1

(c)  ∫__0^(1/2^(1/2)) 1/(1 - x^2)^(1/2) dx =[sin^(-1)(x)   ] _ ( 0)^(1/2^(1/2)) = sin^(-1)(1/2^(1/2)) - sin^(-1)(0) = π/4 - 0 = π/4


(2)  Find the area of the region under the graph of   y = 1/x^2 between   x = 1  and   x = 2.

∫_1^21/x^2 dx =∫_1^2x^(-2) dx =[x^(-1)/-1] _1^2=[-1/x] _1^2 = -1/2 - -1/1 = 1/2square unit of area


(3) Suppose F(x) = ∫_1^x (t + 3)^(1/2)/cos(4t) dt .   Find F ' (x)

By F.T.C II, F ' (x) = (x + 3)^(1/2)/cos(4x)