Evaluate the Integral

Problem

(∫_0^11)(√(,121−x2)*d(x))

Solution

1. Identify the integral as representing the area of a specific geometric shape. The function y=√(,121−x2) represents the upper half of a circle centered at the origin with radius r=√(,121)=11

2. Determine the region of integration. The limits are from x=0 to x=11 which corresponds to the first quadrant of the circle.

3. Apply the area formula for a circle. Since the region is one-quarter of a full circle, the area A is given by 1/4*π*r2

4. Substitute the radius r=11 into the formula.

A=1/4*π*(11)2

5. Simplify the expression to find the final value.

A=(121*π)/4

Final Answer

(∫_0^11)(√(,121−x2)*d(x))=(121*π)/4

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