Question

To calculate the volume of the described solid (parabola).

Short answer

The volume of the parabola is\(2\).

Step-by-step solution

Given data

The base of the solid region is enclosed by the parabola \(y = 1 - {x^2}\) and the \(x\)-axis.

The cross-sections perpendicular to the \(y\)-axis are squares.

Concept used of Volume

Volumeis a scalar quantity expressed in the amount of three-dimensional space enclosed by a closed surface.

Solve to find the volume

Find the base length \((2x)\) of the square cross section as follows.

\(\begin{aligned}{}y = 1 - {x^2}\\{x^2} = 1 - y\\x = \pm \sqrt {1 - y} \\2x = 2\sqrt {1 - y} \end{aligned}\)

The cross-section of the parabola is as shown in Figure 1.

The region lies between \(a = 0\) and \(b = 1\).

The expression to find the volume of the solid region as shown below.

\(V = \int_a^b A (y)dy\) ….(1)

Find the area of square by the use of the relation as shown below.

\(\begin{aligned}{}A(y) = {(2x)^2}\\ = {(2\sqrt {1 - y} )^2}\\ = 4(1 - y)\end{aligned}\)