Use geometry to evaluate the definite integral.
[tex]\int_{-9}^0 \sqrt{81-x^2} d x[/tex]
Answer (2)
Recognize the integral as the area of a quarter circle.
Calculate the area of the full circle with radius 9: A = π ( 9 2 ) = 81 π .
Divide the full circle area by 4 to get the quarter circle area: 4 81 π .
The definite integral evaluates to 4 81 π .
Explanation
Problem Analysis We are asked to evaluate the definite integral ∫ − 9 0 81 − x 2 d x using geometry.
Geometric Interpretation The integrand y = 81 − x 2 represents the upper half of a circle centered at the origin with radius 9, since x 2 + y 2 = 81 .
Limits of Integration The limits of integration are from x = − 9 to x = 0 . This corresponds to the area of the quarter circle in the second quadrant.
Area of the Full Circle The area of a full circle with radius r = 9 is given by A = π r 2 = π ( 9 2 ) = 81 π .
Area of the Quarter Circle Since we are only interested in the quarter circle in the second quadrant, we divide the area of the full circle by 4: 4 81 π .
Final Result Therefore, the value of the definite integral is 4 81 π .
Examples
Imagine you are designing a curved window for a building, and the shape of the window follows the curve described by the function y = 81 − x 2 from x = − 9 to x = 0 . The integral ∫ − 9 0 81 − x 2 d x calculates the area of this window. Knowing this area is crucial for ordering the correct amount of glass and estimating the cost of the window.
The definite integral ∫ − 9 0 81 − x 2 d x represents the area of a quarter circle with radius 9. The area of a full circle is 81 π , so the area of the quarter circle is 4 81 π . Thus, the integral evaluates to 4 81 π .
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