(873, #21) a) Express the volume of the wedge in the first octant that is cut from the cylinder y² + z² = 1 by the planes y = x and x = 1 and as a triple integral.

b)Use either the Table of integrals (on the back end-papers) or a computer algebra system to find the exact value of the triple integral in part (a).

Solution: The planes y = x, x = 1, and the cylinder y² + z² = 1, all restricted to the first octant, are sketched below.

Note that if y is fixed then x varies from y to 1 and z = sqrt(1 - y²). Thus the volume in question is equal to the tripel integral:

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Neither a table of integrals nor Maple is needed to carry out the integration. We have, first

Then

Finally,

Thus the integral is equal to Pi/4 - 1/3.

by: gm