Evaluate the Integral

Problem

(∫_0^1)(√(,1−x2)*d(x))

Solution

1. Identify the integral as representing the area of a specific geometric shape. The function y=√(,1−x2) describes the upper half of a circle centered at the origin with radius r=1

2. Determine the region of integration. The limits from x=0 to x=1 correspond to the portion of the circle in the first quadrant.

3. Apply the area formula for a circle. Since the region is exactly one-quarter of a full circle with radius 1 the area is 1/4*π*r2

4. Substitute the radius r=1 into the area formula to find the value of the definite integral.

5. Calculate the final result.

Area=1/4*π*(1)2

Area=π/4

Final Answer

(∫_0^1)(√(,1−x2)*d(x))=π/4

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