Chapter 15 - Section 15.2 - Double Integrals over General Regions - 15.2 Exercise - Page 1008: 6

$$\frac{2\left(e+1\right)^{\frac{3}{2}}-4\sqrt{2}}{3}$$

Given $$\int_{0}^{1}\int_{0}^{e^v}\sqrt{1+e^v}dwdv$$ Since \begin{align*} \int_{0}^{1}\int_{0}^{e^v}\sqrt{1+e^v}dwdv&=\int_{0}^{1}\sqrt{1+e^v} [w]_{0}^{e^v}dv\\ &=\int_{0}^{1} e^v\sqrt{1+e^v} dv\\ &=\frac{2}{3}(1+e^v)^{3/2}\bigg|_{0}^{1}\\ &=\frac{2\left(e+1\right)^{\frac{3}{2}}-4\sqrt{2}}{3} \end{align*}