Question

Estimate the area under the curve f(x) = 16 – x^2 from x = 0 to x = 3 by using three inscribed (under the curve) rectangles. Answer to the nearest integer.

Answer 1

The inscribed rectangles will have width 1. The x-values that result in inscribed (as opposed to circumscribed) rectangles are {1, 2, 3}. Subst. these values into the function f(x) = 16 - x^2 results in the following y-values: {15, 12, 7}.

The three inscribed rectangles have the following areas: 1{15, 12, 7}, since the width of each rectangle is 1. Thus, the areas are {15, 12, 7}.

To estimate the area under the curve, sum up the areas of these rectangles:

15+12+7. Answer: 34 square units. Note that the exact area, obtained through integral calculus, is 39 square units.