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 |
b
∫
a
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
∫
a
bf(x)dx=[g(x)]ab=g(b)−g(a) ∫√a2−x2dx=‌
x
2
√a2−x2+‌
a2
2
sin−1‌
x
a
+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 ‌=