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 = 2 sin x, y = sin 2x, 0 ≤ x ≤ π

Accepted Answer

Welcome back everyone. Find the area of the shaded region. A 2 square units, B 1 square unit, C 4 square units, and D 3 square units. For this problem we're going to define the area as the integral from A to B of F of X minus G of XDX. Remember that F of X is the upper curve and G of X is the lower curve. In this case, our region is between x equals 0 and x equals 5 divided by 2. So those are the limits of integration A and B respectively. The upper curve is the red one. It is y equals 4 sin x, so that's going to be f of x, and the lower curve is the blue one, right, which is g of x. So we can now set up the integral. Area equals integral from 02. I divided by 24 sin x minus 2 sin 2 x dx. Well done, and now let's go ahead and integrate. Let's remember that the integral of sine is negative cosine, so for the first part, we get -4 cosine x. And for the second part, we subtract. 2, the integral of sin 2 x is going to be negative cosine 2 x divided by 2 according to the reverse chain rule, right? And we are going to evaluate this expression between 0 and 5 divided by 2. Let's go ahead and simplify. This is going to be. 2 divided by 2 gives us 1, right? We can cancel out the negative signs to get cosine of 2 x minus 4 cosine x. From 0 to pi divided by 2. So now let's substitute the limits of integration. We're going to get cosine of. 2 multiplied by pi divided by 2 is cosine of pi minus 4 cosine of pi divided by 2. And we're going to subtract cosine of 0 minus 4 cosine of 0. So now let's evaluate each part. We have cosine of pi, which is -1. Minus for a cosine of pi divided by 2, which is 0. And then we're subtracting. Cosine of 0, so basically we're subtracting 1 minus 4 cosine of 0, right? minus -4 cosine of 0, so we're basically adding 4 multiplied by 1. And basically we get -1 minus 1 + 4, which is 2 square units, and this corresponds to the answer choice A. Thank you for watching.

Find the areas of the regions enclosed by the curves and lines in Exercises 15–26.y = 2 sin x, y = sin 2x, 0 ≤ x ≤ π

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

y = 2 sin x, y = sin 2x, 0 ≤ x ≤ π