Answer
Area $=\displaystyle \int_{0}^{2}y^{3}dy$
Work Step by Step
Th. 4.5 The Definite lntegral as the Area of a Region
If $f$ is continuous and nonnegative on the closed interval $[a, b]$,
then the area of the region bounded by
the graph of $f$,
the x-axis, and
the vertical lines $x=a$ and $x=b$ is
Area $=\displaystyle \int_{a}^{b}f(x)dx$.
(See Figure 4.22.)
----------------------
Here, x is a function of y.
$x=f(y)$
(the function values are read off the x-axis)
(see the "dummy variable" discussion under fig.4.24.)
The variable of integration here is y,
(on the y-axis,) for $y\in[a, b]$ = $[0,2]$
$f(y)=y^{3}$ is continuous and nonnegative on the closed interval,
(just the setup:)
Area $=\displaystyle \int_{0}^{2}y^{3}dy$
