MATH 021 Midterm: math21_2006f_exam3_soln
Document Summary
Uc merced, fall 2006: solve the following unrelated problems (a) (10 points) write down, but do not evaluate, a riemann sum to estimate the de nite integral. 0 ptan(x) dx using n = 3 subintervals and right-endpoints. Solutions: each subinterval has the same length and the endpoints of these intervals are. Using left endpoints 0, 1/3, and 2/3, we get that the heights for three rectangles are. Hence, the riemann sum is ptan(0), ptan(1/3), andptan(2/3). 3ptan(2/3). (b) (10 points) calculate the exact area enclosed between the x axis and the parabola y = 1 x2. Solutions: it helps to sketch a graph of the parabola as follows. So the area enclosed between the x axis and the parabola is. =(cid:20)1 (1 x2) dx = x . Or, because 1 x2 is an even function, a slightly simpler calculation goes as. 3 (cid:12)(cid:12)(cid:12) (1 x2) dx = 2z 1 (1 x2) dx == 2(cid:20)x .
