   Chapter 15, Problem 36RE

Chapter
Section
Textbook Problem

Find the volume of the given solid.36. Under the surface z = x2y and above the triangle in the xy-plane with vertices (1,0), (2, 1), and (4, 0)

To determine

To find: The volume of the given solid.

Explanation

Given:

The region E lies under the surface z=x2y and above the triangle in the xy-plane with vertices (1,0),(2,1),(4,0) .

Formula used:

The volume of the given solid is given by, V=EdV .

Calculation:

From the given conditions, it is observed that the equations of the sides of the triangle y+1 and 42y . Then, the volume of the given solid is,

V=EdV=01y+142y0x2ydzdxdy

Integrate with respect to z and apply the limit.

V=01y+142y[z]0x2ydxdy=01y+142y[x2y0]dxdy=01y+142yx2ydxdy

Integrate with respect to y and apply the limit.

V=01[x3y3]y+142ydy=01[y(42y)33y(y+1)33]dy=01[y(6496y+48y28y3)3y(y3

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