Question

Find the volume of the described solid s. the solid s is bounded by circles that are perpendicular to the x-axis, intersect the x-axis, and have centers on the parabola y = 1 2 (1 − x2), −1 ≤ x ≤ 1.

Answer 1

Answer:

$\bf\implies Volume = \frac{4\cdot\pi}{15}\textbf{ cubic units}$

Step-by-step explanation:

For better understanding of the solution, see the attached diagram of the problem :

The solid thus formed is a semicircle

$\text{The centre of the solid is at parabola } y = \frac{1}{2}\cdot (1-x^2)\\\\\implies \text{Radius of the semicircle = }\frac{1}{4}\cdot (1-x^2)\\\\\text{Now, Volume of the solid = }\int\limits^{1}_{-1} {\pi\times radius^2} \, dx \\\\Volume=\int\limits^{1}_{-1} {\pi\times \frac{1}{4}(1-x^2)^2} \, dx\\\\\implies Volume=\frac{\pi}{4}\times \int\limits^{1}_{-1} {(1-x^2)^2} \, dx\\\\\text{On solving the above integration,}\\\\\bf\implies Volume = \frac{4\cdot\pi}{15}\textbf{ cubic units}$