Question

If the area under the curve of f(x) = 25 – x2 from x = 0 to x = 5 is estimated using five approximating rectangles and left 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

B. overestimate.

Explanation:

Let's find the answer, but you can check the attached file for clarity.

First, we need to understand that the given equation
`f(x) = 25 – x^2` has a negative slope between x = 0 and x = 5, this means that for 2 different values of 'x', for example x=2 and x=3, the lowest value of 'x' (x=2) gives a higher f(x) value, which means f(x=2)>f(x=3).

Because we are using the left endpoint for the rectangles, using the same two points (x=2 and x=3), we can observe in the attached file that the left corner of the rectangle (x=2) will be on the curve (f(x)), whereas the right corner of the rectangle (x=3) will be above the curve (f(x)).

In conclusion, because you will be adding extra area per rectangle, the estimated area under the curve will be overestimated, so the answer is B. overestimate.