Question

If the area under the curve of f(x) = x^2 + 2 from x = 1 to x = 6 is estimated using five approximating rectangles and right endpoints, will the estimate be an underestimate or overestimate?

A. Underestimate
B. Overestimate
C. The area will be exact
D. The area cannot be estimated with just five rectangles

Answer 1

Answer: Overestimate.

Explanation:

f(x) = x^2 + 2

This parabola faces upward (coefficient of x^2 is positive), with minimum at df(x)/dx = f'(x) = 2x = 0, x = 0.

We are asked about area under the parabola in a region where it is increasing as x increases, namely, from x=1 to x=6. So the right endpoint is always greater than the left endpoint. The entire mismatch between the rectangle and the parabola appears _above_ the curve, so the answer must be "overestimate".

To check the answer, we compute the exact area and compare with the approximation: It is well known that the area is F(6)-F(1) with F(x) any function such that dF(x)/dx = F'(x) = f(x), for example, F(x) = x^3/3 + 2x.

F(6) = 6×6×6/3 + 2×6 = 72+12 = 84

F(1) = 1×1×1/3+2 = 7/3

Exact area is 84-7/3 = 81+2/3

Now, the appropriate area using right endpoints with 5 rectangles is

1×f(2) + 1×f(3) + 1×f(4) + 1×f(5) + 1×f(6)

= 6 + 11 + 18 + 27 + 38 = 100.

Since this is greater than 81+2/3, "overestimate" is indeed correct. ✔

Answer 2

(B) Overestimate

Explanation:

This parabola is convex (open up) and the interval of estimation interrogates the curve in a range where the function is increasing.

If the rectangles match the function value with their upper-right corners, the upper rectangle side is never below the actual curve. This is true for all five (or any number of) rectangles in the interval. Therefore the estimated area will be strictly an overestimate.