Evaluate the Integral

Problem

(∫_−2^2)(√(,4−x2)*d(x))

Solution

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

2. Determine the limits of integration. The integral from x=−2 to x=2 covers the entire upper semicircle.

3. Apply the formula for the area of a semicircle, which is A=1/2*π*r2

4. Substitute the radius r=2 into the area formula.

5. Calculate the final value.

r=2

A=1/2*π*(2)2

A=1/2*π*(4)

A=2*π

Final Answer

(∫_−2^2)(√(,4−x2)*d(x))=2*π

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