   Chapter 7.7, Problem 2E

Chapter
Section
Textbook Problem

# The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate ∫ 0 2 f ( x )   d x , where f is the function whose graph is shown. The estimates were, 0.7811, 0.8675, 0.8632, and 0.9540, and the same number of subintervals were used in each case.(a) Which rule produced which estimate?(b) Between which two approximations does the true value of ∫ 0 2 f ( x )   d x lie? To determine

(a)

To find: Which rule produced which estimate?

Explanation

Given:

The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate 02f(x)dx where f is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, 0.9540, and the same number of sub-intervals were used in each case.

Calculation:

We know for the given function as can be observed from the graph:

RnTnMnLx

f is decreasing and concave up (it can be observed that rate of decrease is decreasing).

Left endpoints: large parts of the upper right corners of the rectangles are outside the region, so that should be the largest estimate 0.9540.

Right endpoints: It should be the smallest estimate because it can be seen that all the rectangles are inside the region with extra space left between the tops of the rectangles and the curve of f

To determine

(b)

To find: Between which two approximations does the true value of 02f(x)dx lie?

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