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Find the volume of the solid that is generated by rotating around the indicated axis the plane region bounded by the fiven curves.

1) y=√x,y=0,x=4; The x-axis
2) y= 1/x, y=0, x=0.1,x=1; the x-axis

3)Find the volume of the ellipsoid generated by rotating around the x-axis the region bounded by the ellipse with equation.
(x/a)^2 + (y/b)^2 =1

4) (a) Find the volume of the unbounded solid generated by rotating the unbounded region around the x-axis. This is the region between the graph of y=e^-x and the x-axis for x >1. (Compute the volume from x=1 to x=b, where b>1. Then find the limit of this volume as b->+∞).
(b) What happens if y= 1/√x instead?

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Solution Summary

The volume of solids and ellipsoid for an unbounded solid are determined.

Solution Preview

(1) By the disc method we get

pi. int._{0 to 4} (sqrt(x))^2 dx = 8 .pi

Using cylindrical shells we get

2.pi int_{0 to 2} y (4 - y^2) dy = 2.pi (8 - 4) = 8.pi

(2) the ...

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