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While investigating ways to approximate definite integrals, it is insightful to compare results to integrals whose exact values we know. To that end, the following sequence of questions centers on .

(a) Use the above applet to compute , the left Riemann sum with three subintervals, of the function on the window of values from to .

(b) Likewise, use the applet to compute and , the right and middle Riemann sums with three subintervals, respectively.

(c) Use the Fundamental Theorem of Calculus to compute the exact value of .

(d) We define the error in an approximation of a definite integral to be the difference between the integral's exact value and the approximation's value. What is the error that results from using ? From ? From ?

(e) Let us develop a new approach to estimating the value of a definite integral known as the Trapezoid Rule. The basic idea is to use trapezoids, rather than rectangles, to estimate the area under a curve. What is the formula for the area of a trapezoid with bases of length and and height ?

Area of Trapizod

(f) Working by hand, estimate the area under on using three subintervals and three corresponding trapezoids. Call this value .

What is the error in this approximation?

How does it compare to the errors you calculated in (d)?

You can earn partial credit on this problem.