Question

To represent the integral function as an area and to calculate the volume of the solid region (circular disk).

Short answer

The integral function represents the area of quarter circle with radius \(r\).

The volume of the solid region is \(V = \frac{1}{2}\pi {r^2}h\).

Step-by-step solution

Given data

The solid is a circular disk with radius \(r\).

The region lies between \(a = - r\) and \(b = r\).

Concept used of Integral function

An integral in mathematics is either a numerical value equal to the area under the graph of a function for some interval or a new function, the derivative of which is the original function.

Solve to find the volume

The integral function for the volume of the circular disk is \(V = 2h\int_0^r {\sqrt {{r^2} - {x^2}} } dx\).

The integral function represents one quarter of the area of a circle with radius \(r\).

Find the volume of the solid region based on the representation of the integral function as shown below.

\(V = 2h \times \frac{1}{4} \times \pi {r^2} = \frac{1}{2}\pi {r^2}h\)

Therefore, the volume of the solid region \(V = \frac{1}{2}\pi {r^2}h\).