Evaluate the Integral

Problem

(∫_−4^4)(√(,16−x2)*d(x))

Solution

1. Identify the geometric interpretation of the integral. The expression y=√(,16−x2) represents the upper half of a circle centered at the origin with a radius r=4

2. Determine the area being calculated. Since the limits of integration are from x=−4 to x=4 the integral represents the area of the entire semicircle.

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

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

A=1/2*π*(4)2

5. Simplify the expression to find the final value.

A=1/2*π*(16)

A=8*π

Final Answer

(∫_−4^4)(√(,16−x2)*d(x))=8*π

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