Mathematics

Problem

A solid has base the region bounded by the graphs of

y=x

and the x-axis on

0\le x\le 2

. At each

x

, the cross-section perpendicular to the x-axis is a rectangle whose height is

x^{2}

. What is the volume of the solid? a)

\dfrac{8}{3}

\dfrac{16}{3}

4

2

Intuitive explanation

The key is recognizing that you're building up a solid by stacking infinitely many thin rectangular slices perpendicular to the x-axis. At each position $x$ , the base of your rectangle is determined by the region between $y=x$ and the x-axis (so width $x$ ), and the height extends upward by $x^2$ . To find total volume, integrate the area of each cross-sectional rectangle across the interval $[0,2]$ .

Intuitive explanation

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