Concept: - The area under the function
y=f(x) from
x=a to
x=b and the
x-axis is given by the definite integral
|f(x)dx|, for curves which are entirely on the same side of the
x-axis in the given range.
- If the curves are on boththe sides of the
x-axis, then we calculate the areas of both the sides separately and add them.
- Definite integral: If
∫f(x)dx=g(x)+C, then
bf(x)dx=[g(x)]ab=g(b)−g(a) ∫√a2−x2dx=√a2−x2+sin−1+C Calculation: Let's first find the points where the curve meets the
x-axis
(y=0).
⇒y=√16−x2=0 ⇒x=±4 Now, sincethe curve
y=√16−x2 is entirely on one side of the
x-axis in the given range
x=−4 to
x=4, we have:
The required area =√42−x2dx =[√42−x2+sin−1]−44 =[√42−42+sin−1]−[√42−(−4)2+sin−1] =8+8 =8π square units.