Use the method of cylindrical shells to find the volume V...

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 6. y = 16x^4, y = 0, x = 1 Sketch the region and a typical shell. Rotating a vertical strip around x = 6 creates a cylinder with radius r = and height h =

Transcript

00:01 For this problem let us begin with the following let r be the region bounded by the following curves first we have y to equal to 16th x to the fourth powered and y here equals zero and finally we have the line x equals one so we have three curves so let us sketch this region so first we are going to have our y -axis then our x -axis here at the bottom the region here is going to be bounded by those given curves so in our case we have y equals to 16 x to the fourth powered so let us sketch that we are going to have something that looks or resembles that of a parabola y equals to 16 x to the fourth powered y co -0 is the x -axis so let's label this as y -co -0 and we are going to have x equals 1 so somewhere over here in the right side we have x equals 1 and the region our region here is going to be the region bounded by the curves let me move this a little bit to the right perfect so here we are going to have our region r let's label this with um capital r and what i want to do is i want to replicate this to have a sense of what is going to happen when i revolve the region r about the axis which is the line x equals 6 is going to be a line to the right side of this at x equals six and let's exaggerate and say that x equal six is this line here so what we're going to do is we're going to imagine taking a slice of the region capital r and revolving this about the line x equal six so imagine that we do that we're going to end up with a circular cylinder as such there we go so pretty much we are revolving this little cylinder about the line x equals six and we are going to have the radius of the cylinder to be let's call this length right here from this to the axis of rotation is going to be little r and in our case little r is going to equal to six minus x in the meanwhile the height of the cylinder the height of the cylinder is going to be this little height right here let's call that h and h is just going to equal to the value of the function at x in our case 16th x to the fourth powered so let's just remember that once again we are revolving the origin about the line x equals six so this is going to be our axis of rotation and the volume to compute the volume we are going to rely on the method of shells so using the shell method the setup for the shell method is let's use capital v for the volume is equal to the integral from a to b of 2 pi times little r times little h times the x.

03:20 So in our case we're going to have the integral.

03:22 The region exists between the x values of 0 and 1...

Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 6. y = 16x^4, y = 0, x = 1 Sketch the region and a typical shell. Rotating a vertical strip around x = 6 creates a cylinder with radius r = and height h =

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