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

`\bf\implies Volume = (4\cdot\pi)/(15)\textbf{ cubic units}`

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 = (1)/(2)\cdot (1-x^2)\\\\\implies \text{Radius of the semicircle = }(1)/(4)\cdot (1-x^2)\\\\\text{Now, Volume of the solid = }\int\limits^(1)_(-1) {\pi* radius^2} \, dx \\\\Volume=\int\limits^(1)_(-1) {\pi* (1)/(4)(1-x^2)^2} \, dx\\\\\implies Volume=(\pi)/(4)* \int\limits^(1)_(-1) {(1-x^2)^2} \, dx\\\\\text{On solving the above integration,}\\\\\bf\implies Volume = (4\cdot\pi)/(15)\textbf{ cubic units}`