Help on calculus please: Area approximation

Question

List x₁, x₂, x₃, x₄ where x_i is the midpoint endpoint of the five equal intervals used to estimate the area under the curve of f(x) between x = 0 and x = 10

Options:

1. 2, 4, 6, 8, 10
2. 1, 3, 5, 7, 9
3. 0, 2, 4, 6, 8
4. 1, 2, 3, 4, 5

Answer 1

First, lets understand that there needs to be 5 equal rectangles that will cover the entire area under the curve between x = 0 and x = 10. Next, we see that the numbers that are given in the options are the midpoints of the rectangles. Knowing these bits of information, let's take a closer look at the given options.

Option 1:

If the first midpoint is x = 2, then the rectangle must be from x = 0 to x = 4, since (0+4)/2 = 2. We see that each rectangle in this option has length of 4. So the next rectangles will be x = 4 to x = 8 and x = 8 to x = 12. By the third rectangle, we already go out of the bounds given to us, which was up to x = 10. So this cannot be the answer.

Option 2:

If the first midpoint it x = 1, then the length of the rectangle must be from x = 0 to x = 2, since (0+2)/2 = 1. We see that each rectangle in this option has length 2. So the next rectangles will be x = 2 to x = 4 (with midpoint x = 3) , then x = 4 to x = 6 (with midpoint x = 5), then x = 6 to x = 8 (with midpoint x = 7) and finally x = 8 to x = 10 (with midpoint x = 9).

This option satisfies the two requirements we had initially which were that there needs to be five equal rectangles and they have to cover the full area from x = 0 to x = 10. Therefore option 2 is the answer.

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