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.