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5. A problem to connect the Riemann sum and the Fundamental Theorem of Calculus: (a) Evaluate...

5. A problem to connect the Riemann sum and the Fundamental Theorem of Calculus:

(a) Evaluate the Riemann sum for f(x) = x 3 + 2 for 0 ≤ x ≤ 3 with five subintervals, taking the sample points to be right endpoints.

(b) Use the formal definition of a definite integral with right endpoints to calculate the value of the integral. Z 3 0 (x 3 + 2) dx.

Note: This is the definition with limn→∞ Xn i=1 f(xi)∆x (

c) Use the Fundamental Theorem of Calculus to check your answer in part (b).

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