Find the areas of the regions enclosed by the curves and lines...

Question

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.

y = sin x, y = x, 0 ≤ x ≤ π/4

Accepted Answer

Welcome back everyone. Find the area of the shaded region a 2 minus pi 2 divided by 12, B 1 minus pi 2 divided by 72, C 1/2 plus pi 2 divided by 72, and d 1/2 minus pi 2 divided by 72. So for this problem we're going to set up an integral. Area is equal to the integral. Now what we're going to do is identify the limits of integration, and based on the image, the region starts at x equals 0, so that's the lower limit of integration and ends at pi divided by 6. So that's the upper limit of integration, and the region is bounded by two curves. So remember that we are subtracting the lower curve from the upper curve, and in this case the upper curve is the red one. It is defined by cosine x. So we're going to take cosine x and we're going to subtract the lower curve, which is y equals x. This is the blue one, right? So we're subtracting x and we're integrating with respect to x. So now that we have the integral we can integrate the integral of cosine is sin x, and the integral of X is going to be x 2 divided by 2 using the power rule, so we get sin x minus x2 divided by 2 evaluated from 0 up to pi divided by 6. And now we are ready to apply the limits of integration, which gives us sin of pi divided by 6 minus. 1/2 multiplied by Pi divided by 6 squared. And we're going to subtract sin of 0 minus 02 divided by 2. The sin of 0 is 00 squared divided by 2 is 0. So we just want to simplify the first part. Sin of pi divided by 6 is equal to 1/2, and we're subtracting 1/2 multiplied by pi square divided by 36. Now simplifying further, we get 1/2 minus 2 divided by 72 square units. Which corresponds to the answer of choice the. Thank you for watching.

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.y = sin x, y = x, 0 ≤ x ≤ π/4

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Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.
y = sin x, y = x, 0 ≤ x ≤ π/4