Evaluate the Integral

Problem

(∫_−5^5)(√(,25−x^2)*d(x))

Solution

1. Identify the geometric interpretation of the integral, which represents the area under the curve y=√(,25−x^2) from x=−5 to x=5

2. Recognize that the equation y=√(,25−x^2) is the upper half of a circle centered at the origin with radius r=5 since x^2+y^2=25

3. Determine the area of this region, which is a semicircle because the limits of integration [−5,5] cover the entire diameter of the circle.

4. Apply the formula for the area of a semicircle, which is A=1/2*π*r^2

5. Substitute the radius r=5 into the area formula.

A=1/2*π*(5)^2

6. Simplify the expression to find the final value.

A=(25*π)/2

Final Answer

(∫_−5^5)(√(,25−x^2)*d(x))=(25*π)/2

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