(a) Estimate the area under the graph of f(x) = 25 - x2 from x = 0 to x = 5 using five approximating rectangle

Question

(a) Estimate the area under the graph of f(x) = 25 - x^2 from x = 0 to x = 5 using five approximating rectangles and right endpoints.
b) Estimate the area using left endpoints.

Answer 1

Since you have to divide the area into five rectangles, you will need rectangles from:

0 to 1, 1 to 2, 2 to 3, 3 to 4, 4 to 5

Since you are using the right endpoints, you need to be using the second value of each division; that is, in the "x to y" examples, you'll be using the y endpoints 1,2,3,4, and 5.

The area of each rectangle will then be 1*f(x), since the width of each rectangle is 1. Therefore, the total sum formula should be as follows (with * representing multiplication).

1 * f(1) + 1*f(2) + 1*f(3) + 1*f(4) + 1*f(5)

Or, quite simply, f(1) + f(2) + f(3) + f(4) + f(5) to solve using the right endpoints. Calculating this, we get

(25 - 1^2) + (25 - 2^2) + (25 - 3^2) + (25 - 4^2) + (25 - 5^2)
(24) + (21) + (16) + (9) + 0 = 70

To calculate using the left endpoints, recall that our intervals for the rectangles were: 0 to 1, 1 to 2, 2 to 3, 3 to 4, and 4 to 5. In this case, we use the LEFT endpoints and take the FIRST value of each interval; that is, 0,1,2,3, and 4. The width of the rectangle doesn't change at all (the width is 1), so our left endpoint summation formula because

1 * f(0) + 1 * f(1) + 1 * f(2) + 1 * f(3) + 1 * f(4)

or, quite simply

f(0) + f(1) + f(2) + f(3) + f(4). We calculated f(1) through f(4) already above, so we'll copy those values. But f(0) = (25 - 0^2) = 25, so our total now becomes:

25 + 24 + 21 + 16 + 9 = 95.

Area using right endpoints = 70
Area using left endpoints = 95

Answer 2

To estimate the area under the graph, you need to figure the height f(x) for each of the points. They want you to use five rectangles so that means each will have a width of 1 unit. If you are using the *right* endpoints, that will be the points x = 1, 2, 3, 4, 5.
f(1) = 25 - 1² = 24
f(2) = 25 - 2² = 21
f(3) = 25 - 3² = 16
f(4) = 25 - 4² = 9
f(5) = 25 - 5² = 0

Since each rectangle will have a width of 1, the areas of the five rectangles are 24, 21, 16, 9 and 0 respectively for a total area of 70 sq. units.

(b) Left endpoints:
This is the same method but you use the left endpoints x = 0, 1, 2, 3 and 4.

f(0) = 25 - 0² = 25
f(1) = 25 - 1² = 24
f(2) = 25 - 2² = 21
f(3) = 25 - 3² = 16
f(4) = 25 - 4² = 9

25 + 24 + 21 + 16 + 9 = 95 sq. units

Answer 3

25 X2

Answer 4

You might have better luck with this question if you posted it in the Science and Mathematics category. The people browsing there may be better able to help you.