Math 20B (Bender)Solutions to First Exam18 October 2000I’ve noted if the problem or a near miss is in the text.1. (10 pts.) EvaluateZ20p4-x2dxby interpreting it as an area.A. (p.383, Ex.4) Squaring and rearrangingy=√4-x2givesx2+y2= 4, a circle ofradius 2 centered at the origin. The integral is the area in the first quadrant and soequals (π22)/4 =π.Since the problem did not ask for an exact answer, you will receive credit for areasonable numerical evaluation.2. (30 pts.) Evaluate the following integrals using the tools discussed in the text.Z(1-x)p2x-x2dxZ20|sinπx|dx.A. (p.426, #26, #39) The substitutionu= 2x-x2converts the first toR12u1/2du=u3/2/3 +Cand so the answer is (2x-x2)3/2/3 +C.The second integral equalsR10sinπx dx-R21sinπx dx. The substitutionu=πxgivesRsinπx dx= (-cosπx)/π+C. Thus the answer is(-cosπ+ cos 0)/π-(-cosπ+ cos 2π)/π= 4/π.3. (30 pts.) Differentiate the functionsF(x) =Zx1p1 +u4duG(x) =Z1x2ln(1-t3)dt.A. Both rely on the Fundamental Theorem of Calculus.
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