Finding surface area
Find the area of the surface genera...

Question

Finding surface area

Find the area of the surface generated by revolving the curve in Exercise 23 about the y-axis.

Accepted Answer

Welcome back everyone. Compute the surface area generated by removing the curve y equals 4/3 x. From x equals 1 to x equals 4 about the y axis. A 4 pi square units, b 9 pi square units, C 16 pi square units, and D 25 pi square units. So for this problem we're going to use the formula for the surface area generated. By revolving a curve about the y axis, it is s equals the integral from a to b of 2 pi multiplied by x multiplied by the square root of. 1 plus the derivative of y with respect to x 2 d x. Now what we're going to do is begin by identifying the derivative of y, that's y. It is the derivative of 4/3 x, which is equal to 4/3. We can factor out the constant, and the derivative of x is 1. Now we can apply the formula. The surface areas is equal to. We can factor out 2 pi because it is a constant and integrate. So the integral is going to be from 1 to 4. This is what the problem suggests from x equals 1 to x equals 4, and we're multiplying x by. square root of 1 + y2, so we're adding 1 and 4/3 squared followed by d x. Notice that we're simply getting another constant, so let's go ahead and simplify the constant. We're going to get 2 pi multiplied by square root of. 4/3 squared can be written as 16 divided by 9, and if we add 1, we can simply rewrite it as 9 divided by 9, right? And this gives us 25 divided by 9. Under the radical. Which is equal to 5 divided by 3, so we can factor out 5/3. And we get 2 pi multiplied by 5/3 integral from 1 to 4 of x dx. This is now an elementary integral we can apply the power rule, and we're going to get. 2 pi multiplied by 5, that's some pi divided by 3. The integral of X is going to be x 2 divided by 2 using the power rule, and we want to evaluate the result from 1 to 4. Notice that we can simplify as well. 10 divided by 2 gives us 5 divided by 3. This is our fraction at the very beginning, and we can cross out 2 in the denominator. Of X 2 divided by 2, so we get 5 pi divided by 3 multiplied by x 2 evaluated from 1 to 4. Let's factor out the result on constant 5 pi divided by 3 and in parentheses evaluate x 2 from 1 to 4, so that'd be 42 minus 12. Which gives us 16 minus 1, and that's 15, right? So we get 5 divided by 3, multiplied by 15. 15 divided by 3 is equal to 5, and 5 pi multiplied by 5 is 25 pi. So the answer is 25 pi square units, which is option D. Thank you for watching.

Finding surface areaFind the area of the surface generated by revolving the curve in Exercise 23 about the y-axis.

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