Mathematics

Mathematics

Problem

A solid extends along the x-axis from x=0x=0 to x=4x=4. For 0≤x≤20\le x\le 2, each cross section perpendicular to the x-axis is a square whose side length is 3x+13x+1 units. For 2≤x≤42\le x\le 4, each cross section is a rectangle whose height is 22 units and whose width is x+1x+1 units. Calculate the exact volume of the solid.
Intuitive explanation

The key to this problem is the cross-sectional area method: volume equals the integral of cross-sectional area along the axis of extension. Since the cross sections change shape at x=2x=2, you'll need to split the integral into two regions and find the area formula for each—a square with side 3x+13x+1 for the first part, and a rectangle with dimensions 22 by x+1x+1 for the second—then add the volumes together.

Intuitive explanation

The key to this problem is the cross-sectional area method: volume equals the integral of cross-sectional area along the axis of extension. Since the cross sections change shape at x=2x=2, you'll need to split the integral into two regions and find the area formula for each—a square with side 3x+13x+1 for the first part, and a rectangle with dimensions 22 by x+1x+1 for the second—then add the volumes together.

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