how do you do the shell method of solid revolving around the x-axis between the curves x=y^4/4-y^2/2; x=y^2/2?

First you need to find the end points of the bounded region. Set the two equations equal to each other and solve for y. Let say that the end points are c and d, (You will actually need to find them) After we have the end points we need to visualize the region so we can explore what our shells will look like. In this case the important features of the shell are the radius of the shell (i.e. distance an abritrary shell is from the axis of rotation) It should be obvious that the radius here is going to be y. Next we need the "height" of the shell. That will be the distance between the two curves that define our region. Once you have that you set up the integral as follows \[\int\limits_{c}^{d}2 \pi y h(y) dy\] Here h(y) represents the expression you'd get for the height of an arbitrary shell as a function of y (right hand curve - left hand curve.)

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