9-34. Shell method Let R be the region bounded by the followin...

Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

Graph showing a region bounded by curves, illustrating the shell method for volume calculation when revolved about the x-axis.

x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

Accepted Answer

Let R be the region bounded by X equals 8 divided by Y2 plus Y cubes, X equals 2/3, and Y equals 1. Find the volume of the satellite generated when R is evolved about the X-axis using the shell method. And were given a graph of the region below. Now, to solve this, it tells us to use the shell method. So we will have V as equals to 2 pi multiplied by the integral from A to B of the radius, multiplied by the height. DY Now, let's go ahead and find all of our pieces of the sin. We want our bounds first. We notice on the Y one is our lower bounds. We'll say A equals 1 in this case. Now we need to find the upper bounds. To find this, we will set our equation 8 divided by Y2 plus Y cubed. Equals to 2/3. This will help us find the way. We get 8 equals 2/3, multiplied by Y2 plus Y cuts. And now we can simplify this. We divide both sides by 2/3. To get 12 equals Y2 plus Y cuts. And finally, We can subtract 12. To get 0 equals. Y2 plus Y cubes minus 12th. Now, we can solve this for Y by using a rational route. But if we solve this, we end up getting the first root of Y equals 2. This will be our upper bound, which means B equals 2. Now we need our radius and our height. In this case, the radius will just be Y. And we can find the height by taking the difference of our rightmost function and our leftmost function. In this case our right must function. Will be 8 divided by Y2 plus Y cubed, and our leftmost function will be 2/3. So now we can plug this into our integral. We get V equals 2 pi multiplied by the integral from 1 to 2 of Y multiplied by 8 divided by Y2 plus Y Q minus 2/3 D Y. Now we can solve this integral. We'll first say then That V equals 2 pi multiplied by the integral. Of 8 Y divided by Y2 plus YQs. And minus 2/3 and the integral of Y D Y. We split up the integral into two separate integrals to solve it easier, and distributed R Y. We didn't get 2 pi, multiplied by the interval from 1 to 2. Of 80 divided by y. Squared multiplied by Y + 1. Minus the integral Of 2/3 Y D Y. We get 2 pie. Multified by the integral. Of 8 divided by Y multiplied by Y plus 1. Minus the integral 2/3s Y, meaning we have two separate intervals to take now. Our first integral on the left is a partial fraction decomposition. So, We can factor out our 8s. Giving us 2 pie Multiplied by 8, multiplied by the integral of 1 divided by Y. Y plus 1. Minus the integral Of 2/31. And we can use partial fraction decomposition for the first integral. 1 divided by Y multiplied by Y plus 1. Gives us a divided by Y plus B divided by Y + 1. Solving this gives us a equals. One And B equals -1, meaning we have. 1 divided by Y. -1 divided by Y + 1. We can then rewrite our inner world again. 2 pi multiplied by 8, multiplied by the integral. Of 1 divided by Y minus 1 divided by Y + 1. And then we have minus the integral. Of 2/3 white Do what? We can now take our antiderivatives to pi. Multiply. By its Natural log of Y minus natural log of Y + 1. From 1 to 2. Minus the integral of 2/3 Y. Which we can simplify. Tickets. 2/3 multiplied. By 1/2 squad from 1 to 2. Now we just need to plug in our values. We get 2 pine multiplied. They heat, which is multiplied. By the natural log of 2 minus the natural log of 3. Minus The natural log of 1 minus the natural log of 2. And then we have our other parts. Which will be minus 2/3 multiplied by 1/2 multiplied by 22. Minus 1/2 multiplied by 1 squared. We can now simplify all of this. We get 2 pay multiplied. By 8 natural luck. Uh 4/3. -1. This would be in cubic units. And this is the final answer to our problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

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9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis

Watch next

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

x = 4 / y + y³,x = 1/√3, and y=1; about the x-axis