# application of triple and double integrals

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26. Evaluate the triple integral, where E is bounded by the planes y = 0, z = 0, x + y = 2 and the cylinder y^2 + z^2 =1 in the first octant.

Find the volume of the given solid

30. Under the surface z = x^2y and above triangle in the xy-plane with vertices (1, 0), (2, 1), and (4, 0).

32. Bounded by the cylinder x^2 + y^2 and the planes z = 0 and y + z = 3.

34. Above the paraboloid z = x2 + y2 and below the half-cone z = sqrt(x^2 + y^2).

40. Use the spherical coordinates to evaluate the following triple integral:

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#### Solution Preview

Please see the attached file for detailed solution.

The solid region is bounded above by and below by z = 0. It lies over the triangular region D in the xy¬-plane bounded by the coordinates axes and the line x + y = 2. Then D is defined by the inequalities:

So

The volume ...

#### Solution Summary

The solution shows how to find the volume of solid using double and triple integrals step-by-step. It also explains how to convert the integrals from rectangular coordinates to spherical coordinates.