Answer to Question #346267 in Calculus for doni

Question

Using shell method, find the volume of R, when it is bounded by √x + √y = √a , x =

0 , y = 0 about the line x = a.

Answer 1

$$\sqrt{x}+\sqrt{y}=\sqrt{a},$$

$$x=(\sqrt{a}-\sqrt{y})^2=a-2\sqrt{a}\sqrt{y}+y$$

$$V=\displaystyle\int_{0}^a2\pi y(a-(a-2\sqrt{a}\sqrt{y}+y))dy$$

$$=2\pi\displaystyle\int_{0}^a(2\sqrt{a}y^{3/2}-y^2)dy$$

$$=2\pi[\dfrac{4\sqrt{a}y^{5/2}}{5}-\dfrac{y^3}{3}]\begin{matrix} a \\ 0 \end{matrix}$$

$$=\dfrac{14\pi a^3}{15} ({units}^3)$$

$\dfrac{14\pi a^3}{15}$ cubic units.