Evaluate the following two integrals by interpreting each in terms of areas

Question

∫_0^1▒√(1−𝑥^2 ) 𝑑𝑥

Answer 1

The integral

∫_0^1√(1−𝑥^2 ) 𝑑𝑥

represents the area under the curve y = √(1 - x^2) from x = 0 to x = 1. This curve represents the upper half of the unit circle centered at the origin. The integral calculates the area between the curve and the x-axis in this region.

Since the integral represents the area under the curve between 0 and 1, and this area is in the shape of a quarter of a circle, the integral evaluates to π/4 or approximately 0.7854.