Evaluate the Integral

Problem

(∫_0^3)(√(,9−x2)*d(x))

Solution

1. Identify the integral as representing the area of a specific geometric shape.

2. Recognize that the function y=√(,9−x2) describes the upper half of a circle centered at the origin with a radius r=3 since x2+y2=9

3. Determine the region of integration, which is from x=0 to x=3

4. Observe that this region corresponds to exactly one-quarter of the area of the full circle.

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

6. Substitute r=3 into the formula.

A=1/4*π*(3)2

7. Simplify the expression to find the final value.

A=(9*π)/4

Final Answer

(∫_0^3)(√(,9−x2)*d(x))=(9*π)/4

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