[Technology Exercise] When solving Exercises 33-40, you may ne...

Question

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.

Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.

y = sin x, 0 ≤ x ≤ π

Accepted Answer

Hello. In this video we are going to consider the curve y is equal to cosine of x that runs from 0 to pi divided by 2. When this curve is revolved about the x axis, it generates a surface. We want to find the area of the surface generated by using the trapezoid rule with 3 subintervals, and we want to round our answer to two decimal places. So, let's go ahead and first define the surface area of the curve. Now, the general surface area of the surface is defined as S is equal to 2 pi multiplied by a definite integral from A to B of Y multiplied by the square root of 1 + Y2 DX. Now here we need to define our boundaries. Now the problem tells us that the curve runs from 0 to pi divided by 2. So our smaller boundary A is going to be 0, and our larger boundary B is going to be pi divided by 2. Now here we are told that the surface is being generated by the curve y is equal to cosine of x. So we know that Y is equal to cosine of x, and if we take the derivative of this function, Y is equal to negative sin of x. So if we were to plug in this information into the surface area formula, we get S is equal to 2 pi multiplied by the definite integral from 0 to pi divided by 2 of cosine of X multiplied by the square root of 1 + negative sin x 2, which is 1 + sin square of X D X. Now, what we want to do in this case is you want to estimate the area of this definite integral by using the trapezoidal rule. Now the trapezoidal rule is defined as the following. It is going to be the width of the subintervals delta x divided by 2, and it will be multiplied by g x0 plus 2 multiplied by g x1. Plus all the numbers leading up to 2 multiplied by g of xn minus 1 plus the final end point g of xn. So here what we need to do is we need to first define the width of the subintervals. Now delta X is defined as b minus a divided by n. Here B minus A represents the area between our intervals or our endpoints, and N represents the number of subintervals we're working with. Now, keep in mind that the problem tells us that we want to work with 3 sub-intervals. So n is going to equal to 3 in this case, and our boundaries b minus A is going to be defined as pi divided by 2 minus 0. And this will be divided by 3. This will all simplify to be delta x equal to pi divided by 6. So now that we have the width of the subintervals, the next thing we need to do is we need to define the endpoints that we will plug into our function. Now, our first endpoint is going to start at our lower interval, which is going to be x0 equal to 0. Then if we add a length of pi divided by 6, we get $$X_1 is $$equal to pi divided by 6. Adding another length of x or pi divided by 6 gives us x 2 is equal to pi divided by 3. And if we add another pi divided by 6, we get x3 is equal to pi divided by 2. So in total, we are going to be working with 4 endpoints for the trapezoidal rule. Now here, our function G of X is going to be defined as the integrand, and that is going to be cosine of x multiplied by the square root of 1 plus sin squared of x. So here, if we plug in the first endpoint to our function. G of 0 is going to have an output of 1. Plugging in the next endpoint, G of pi divided by 6 will give us an output of square root 15 divided by 4. G of pi divided by 3 will give us a value of a square root 7 divided by 4. And g of pi divided by 2 will give us an output of 0. So now if we plug this in to the trapezoidal rule, the trapezoidal rule is going to be defined. As pi divided by 6. Divided by 2 multiplied by g of 0, which is 1, + 2 multiplied by g of pi divided by 6, which is square root 15 divided by 4, plus 2 multiplied by g of pi divided by 3, which is square root 7 divided by 4, plus g of pi divided by 2, which is 0. Now, with the help of a calculator, this is going to approximate to give us the value of 1.115099. Now, this is going to be the estimated value of the integral for our surface area, but keep in mind that our surface area is 2 pi multiplied by the estimated value of the integral. So for our final step, we will take the estimated value of the integral by the trapezoid rule, and the surface area is going to be 2 pi multiplied by 1.115099. And if we were to plug this into a calculator, this is going to have an approximate value of 7.01, which is going to be the solution to the problem. And the overall solution to this problem is going to be option D. So I hope this video helps you in understanding on how to approach this problem, and we will go ahead and see you all in the next video.

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.y = sin x, 0 ≤ x ≤ π

Watch next

[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π