Evaluate the Integral

Problem

(∫_−3^3)(√(,9−x^2)*d(x))

Solution

1. Identify the geometric interpretation of the integral, which represents the area under the curve y=√(,9−x^2) from x=−3 to x=3

2. Recognize that the equation y=√(,9−x^2) describes the upper half of a circle centered at the origin with a radius r=3 since x^2+y^2=9

3. Determine the area of this region, which is a semicircle with radius r=3

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

5. Substitute r=3 into the area formula to calculate the value of the integral.

6. Calculate the final value: 1/2*π*(3)^2=(9*π)/2

Final Answer

(∫_−3^3)(√(,9−x^2)*d(x))=(9*π)/2

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