The Fundamental Theorem Another Way: Let h be the square root function h(x) = x1/2. Let

Chapter 5, Problem 8

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The Fundamental Theorem Another Way: Let h be the square root function h(x) = x1/2. Let Pbe the region under the graph of h from x = 4to x = 9 (Figure 5-6d).Figure 5-6da. Evaluate the Riemann sum R10 for P, pickingsample points at the midpoints of thesubintervals. Dont round your answer.b. Let u be a value of x in the interval [4, 9].Let A(u) be the area of the portion of theregion from x = 4 to x = u (Figure 5-6e). Letu be a small change in u. The area of thestrip from x = u to x = u + u equalsA(u + u) A(u). Explain why this area isbetween h(u) u and h(u + u) u. Writeyour result as a three-member inequality.c. Use the inequality you found in part b toprove that dA/du = h(u). This equation iscalled a differential equation.d. Multiply both sides of the differentialequation given in part c by du. Then take theindefinite integral of both sides. Find theconstant of integration by observing thatA(4) must equal zero.e. Find the area of region P by evaluating A(9).Explain why your answer to part a isconsistent with your answer to this problem.