Chapter 14, Problem 1RE

To determine

To calculate: The integral ${\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}$.

Answer to Problem 1RE

Solution: The value of the integral ${\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}$ is $\underset{\_}{4x^5}$.

Explanation of Solution

Given: The integral, ${\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}$.

Formula used: ${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$.

Calculation:

$\begin{array}{c}{\displaystyle \underset{0}{\overset{2x}{\int }}x{y}^{3}dy}=x{\left[\frac{y^4}{4}\right]}_0^{2x}\\ =x\left[\frac{{\left(2x\right)}^4}{4}-0\right]\\ =4x^5\end{array}$