Chapter 16 - Multiple Integration - 16.1 Integration in Two Variables - Exercises - Page 846: 25

$$84$$

\begin{aligned} \int_{2}^{6} \int_{1}^{4} x^{2} d x d y &=\int_{2}^{6} \int_{1}^{4} x^{2} \cdot 1 d x d y\\ &=\left(\int_{1}^{4} x^{2} d x\right)\left(\int_{2}^{6} 1 dy\right)\\ &=\left(\frac{x^3}{3}\bigg|_{1}^{4}\right) \left(y\bigg|_{2}^{6}\right)\\ &=\left(\frac{4^{3}}{3}-\frac{1^{3}}{3}\right)(6-2)=21 \cdot 4=84 \end{aligned}