Write a geometric argument that justifies the following statement:

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Solution:

How do I know that these expressions relate to the region rotated about the x-axis? The first is clearly the disc method, but the radius (the function) is squared, thus removing the square root and leaving behind the function x1. Clearly, the x-boundaries are 1 and 5, and are listed as the limits of integration. If one were to find this same volume using the shell method, the result is the right hand formula. Remember that in the shELL method, we use paraLLEL rectangles (to the axis of rotation), which in this case are horizontal. Thus, we must use horizontal rectangles to find the volume.

Now, to find the length of a representative rectangle (which is horizontal and thus must use y's), we subtract the left boundary of the region (the function) from the right boundary (the line x = 5). Don't forget that my function must be solved for x so that it contains all y's for this shell method. Hence we get the distance from the axis of rotation equals y and the length of each rectangle is 5-(y²+1). The product of these and 2 pi is integrated using the y-boundaries, 0 and 2. To geth these values, simply plug the x-boundaries into the function to get the corresponding y-boundary.

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