what is the appropriate integral to be used in finding the volume generated by revolving about the x-axis the first quadrant area bounded by x+2y=2 and the coordinate axes?

Question

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"what is the appropriate integral to be used in finding the volume generated by revolving about the x-axis the first quadrant area bounded by x+2y=2 and the coordinate axes"

Answer 1

We are going to use the disk method. The body of rotation will be sliced to thin disks perpendicular to the X-axis with the thickness of dx and areas of π y² .

Thus the volume of each disk is π y² dx, and we need to integrate over all values of x (inside the 2D shape which is being rotated).

The line x + 2y = 2 crosses the X axis at x = 2, therefore 2 is the upper bound of the integral.

Now we need to solve for y

y = 1 - x/2

The integral will be as follows

from x = 0 to x = 2 Integral of π y² dx =

from x = 0 to x = 2 Integral of π ( 1 - x/2 )² dx