Consider the region in the first quadrant bounded by the x and y axes, the vertical line x=3 and the curve y = 1 / (x squared + 3). Determine the volume of the solid by rotating this region about the x-axis. Now that is the first part.
I then have to find the coordinates of the centroid of the solid by rotating this region about the x-axis.
I used the disk method to find the volume. A disk of thickness and height y, rotated about the x-axis gives a volume . Since y = 1/(x squared + 3), the volume required is
The exprression (x^2+3) should make you think of a tangent substitution, so let then , and the integrand becomes
Integrating the cosine squared is accomplished by using the identity that
After some algebra and arithmetic I got the volume to be
I hope this helps,
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