Help on calculus please: Area approximation

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?

Options:

1. Underestimate
2. Overestimate
3. The area will be exact
4. The area cannot be estimated with just five rectangles

Answer 1

This problem will be better explained using a diagram but I will try my best here.

The curve of f(x) = 25 - x^2 is concave down i.e it looks like an upside down "U" shape. If we draw five rectangles using left end points, we see that the top right part of each rectangle is above the curve. This means that the area calculated will be an overestimate which is option B.

Hope I was able to help. Feel free to reach out with further questions!