MATH 181 Study Guide - Final Guide: Trapezoidal Rule, Trigonometric Substitution, List Of Trigonometric Identities
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Problem 1 solution: use the trapezoid rule with n = 2 to estimate the arc-length of the curve y = sin x between x = 0 and x = . Solution: the arclength is: dx a s1 +(cid:18) dy dx(cid:19)2. We now use the trapezoid rule with n = 2 to estimate the value of the integral. 2 hf (0) + 2f(cid:16) where f (x) = 1 + cos2 x and the value of x is: 2 (cid:20) 1 + cos2 0 + 2r1 + cos2. 2. (a) let r be the region between y = 1. 1+x2 and the x-axis with x 0. If so, what is the area? (b) let s be the solid obtained by revolving r around the y-axis. Solution: (a) the area of r is given by the improper integral: We evaluate the integral by turning it into a limit calculation. The integral has a simple antiderivative so its value is:
