Evaluate the Integral

Problem

(∫_0^2)(√(,4−x2)*d(x))

Solution

1. Identify the integral as representing the area of a specific geometric shape. The function y=√(,4−x2) describes the upper half of a circle centered at the origin with a radius r=2

2. Determine the region of integration. The limits of integration are from x=0 to x=2 which corresponds to the portion of the circle in the first quadrant.

3. Recognize that this region is exactly one-quarter of the area of a full circle with radius 2

4. Apply the formula for the area of a circle, A=π*r2 and divide by 4

A=1/4*π*(2)2

5. Simplify the expression to find the final value.

A=1/4*π*(4)

A=π

Final Answer

(∫_0^2)(√(,4−x2)*d(x))=π

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