Determine the area under the curve y = √a^2 - x^2, included between the lines x=0 and x=4.

Question

Determine the area under the curve y = \(\sqrt{a^2 - x^2}\), included between the lines x=0 and x=4.

Answer 1

Given the boundaries of the area to be found are, 

• The curve y = \(\sqrt{a^2-x^2}\)

• x = 0 (y-axis) 

• x = 4 (a line parallel toy-axis)

Here the curve, y =  \(\sqrt{a^2-x^2}\), can be re-written as

y² = a² - x²

x² + y² = a² .....(1)

This equation (1) represents a circle equation with (0,0) as center and, a units as radius. As x and y have even powers, the given curve will be about the x-axis and y-axis.

As per the given boundaries, 

• The curve \(\sqrt{a^2-x^2}\) , is a curve with vertex at (0,0). 

• x=4 is parallel toy-axis at 4 units away from the y-axis. (but this might not really effect the boundaries as the value of ‘a’ in the equation is unknown.) 

• x=0 is the y-axis. 

Area of the required region = Area of OBC.

[sin ^-1(1) = 90° and sin ^-1(0) = 0°] 

The Area of the required region = \(\frac{\pi a^2}{4}\) sq.units