a estimate the area under the graph of fx 4 cosx from x 0 to x 2 using four approximating rectan

A estimate the area under the graph of fx 4 cosx from x 0 to...

(a) Estimate the area under the graph of f(x) = 4 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints. (Round your answers to four decimal places.) R4 = (b) Is your estimate an underestimate or overestimate? (c) Repeat part (a) using left endpoints. L4 = (d) Is your estimate an underestimate or overestimate?

Transcript

00:01 In this problem, we are provided with the function f of x which equals to 4 times cos of x and x varies from 0 up to pi over 2.

00:16 We are asked to find out the area under the graph.

00:20 In subpart a, we are asked to use the right hand sum.

00:23 That is, we are asked to evaluate r4.

00:26 So let us first find out del x.

00:28 Del x equals to b minus a divided by n.

00:32 So substituting b which is pi over 2 minus a which is 0 divided by n which is 4.

00:39 We obtain pi over 8.

00:41 Now let us consider the intervals.

00:44 We have 0 to pi over 8.

00:47 Pi over 8 up to pi over 4.

00:54 Pi over 4 up to 3 times pi over 8 and 3 times pi over 8.

01:01 Over 8 up to pi over 2.

01:04 Now we need to evaluate the function at the right endpoints.

01:10 So we have f of pi over 8 to be equal to 3 .696 approximately.

01:16 F of pi over 4 equals to 2 .8 to 8 to 8 approximately f of 3 pi over 8 equals to 1 .531 approximately and f of pi over 2 .531 approximately and f of pi over 2 .5.

01:31 Equals to 0 since cost of pi over 2 is 0.

01:34 So let us consider the formula of r4 now.

01:38 R4 equals to del x times f of x1 plus f of x2 plus f of x3 plus f of x4.

01:49 Substituting the values we have pi over 8 times 3 .696 plus 2 .828 plus 1 .531 plus 1 .531 plus 0.

02:02 Adding these up and multiplying and dividing by pi and 8 respectively, we obtain 3 .1632...

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