Answer
$$\frac{243}{4}$$
Work Step by Step
Given
$$y=\sqrt[3]{x}, \ \
0 \leqslant x \leqslant 27$$
From the graph of the region, we can observe that area of the bounded region approximately equal to $\frac{3}{4}$ area of the rectangle with width 3 and length 27
$$\text{Area} \approx \frac{3}{4} (3)(27)=60.75 $$
Now we use integration
\begin{aligned} \text{Area}&= \int_0^{27}\sqrt[3]{x}dx\\
&= \frac{3}{4}x^{4/3}\bigg|_{0}^{27} \\
&=\frac{3}{4}(27)^{4/3}\\
&= \frac{243}{4}=60.75 \end{aligned}