If f is integrable on a, b), then \int_{a}^{b} f(x) d x=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} f\left(x_{i}\right) \Delta x \text { where } \Delta x=\frac{b \cdot a}{n} \text { and

} x_{i}=a+i \Delta x \text { Use this definition of the intergral to evaluate } \int_{0}^{1} x^{3}+2 x^{2} d x \text {. } First of all, we have \int_{0}^{1} x^{3}+2 x^{2} d x=\lim _{n \rightarrow \infty} \sum_{i=1}^{n} Evaluating the sum gives \int_{0}^{1} x^{3}+2 x^{2} d x=\lim _{n \rightarrow \infty} Evaluating the limit gives \int_{0}^{1} x^{3}+2 x^{2} d x=