Chapter 4 - Integrals - 4.3 The Fundamental Theorem of Calculus - 4.3 Exercises - Page 328: 40

$$\frac{1}{4}$$

Since the region bounded by $$ y=x^{3}, \quad y=0, \quad x=1 $$ and $$ x^3\geq 0 \ \ \ \text{for}\ \ \ 0\leq x\leq 1 $$ Then area is given by \begin{aligned} \text { Area }&=\int_{0}^{1} x^3 d x\\ &=\left[\frac{1}{4} x^{4}\right]_{0}^{1}\\ &=\frac{1}{4}\\ \end{aligned}