88. The region in Exercise 87 is revolved about the x-axis to ...

Question

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

a. Find the volume of the solid.

Accepted Answer

Welcome back everyone. Consider the region in the first quadrant bounded above by y equals cosequent x below by y equals cotangent x. And between the vertical lines x equals 0 and x equals pi divided by 2, find the volume of the solid formed when this region is revolved about the x axis, a pi square divided by 6, b pi 2 divided by 2, c pi divided by 3, and d 2 pi divided by 3. So for this problem we can use the washer method. Remember that volume equals pi multiplied by the integral from a to b of the outer radius R2 minus the inner radius lowercase r2. We're going to integrate with respect to x, right? So now let's remember that. The outer radius R is basically the curve that is above the region. So that's cosequent x, meaning we're going to take cosequent squared of x and we're going to subtract the curve that is below. So that's cotangent x and we have to square it, right? So that's cotangent squared of x. Now, what about the limits of integration? Well, it says. Between the vertical lines x equals 0 and x equals pi divided by 2, so those are the limits of integration. Well then we have our integral set up, so we get pi multiplied by the integral from 0 to pi divided by 2. And let's remember that according to trigonometry, code sequin 2 x minus cotangent square of x is equal to 1. So we're integrating 1 dx or simply dx. The integral of dx is equal to x, so we get pi multiplied by x, evaluated from 0 to pi divided by 2. And this is where we apply the fundamental theorem of calculus part 2. We can factor out pi and in parent evaluate pi divided by 2 minus 0, which gives us pi square divided by 2 cubic units, and this corresponds to the answer of choice b. Thank you for watching.

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.a. Find the volume of the solid.

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88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.