Given curve: y=16−x2
y2=16−x2
x2+y2=16
This is the equation of a circle with center at origin (0,0) and radius =16=4 units.
The original equation is y=16−x2 with the square root.
Since square root gives only positive values, y≥0
This represents only the upper half of the circle (above the x-axis).
When y=0:
0=16−x2
0=16−x2
x2=16
x=±4
The curve touches the x-axis at x=−4 and x=4.
The region bounded by the curve and x-axis is a semicircle (upper half).
Area of semicircle =21×πr2
Where radius r=4:
Area =21×π×(4)2
Area =21×π×16
Area =8π sq. units
Therefore, the area of the region is 8π square units.