Question

(A) Estimate the area under the graph of f(x)=55−x^2 from x=−3 to x=3 using three approximating rectangles and right endpoints.

Estimate =

(B) Repeat part (A) using left endpoints.

Estimate =

(C) Repeat part (A) using midpoints.

Estimate =

Answer 1

Okay, see the diagram i've attached. The rectangles show only the space taken up on the x-axis; the heights will differ. For part A, you'll use the height of the curve at the right edge of each rectangle: x = -1, x = 1, x = 3. The height of the curve is simply f(x) at that point, f(-1), f(1), f(3). You'll multiply the height by the width of the rectangle, which is 2. Summing the area of the three rectangles gives you an estimate of the area under the curve.

For part B, you'll use the height at the left endpoint, so f(-3), f(-1), f(1).

For part C, you use the midpoint, so f(-2), f(0), f(2).

If you use enough thin rectangles, you get a very good measurement of the area under the curve.