Anonymous
Anonymous asked in Science & MathematicsMathematics · 9 years ago

Evaluate the following integral by interpreting it in terms of areas?

2

{ square root of 4- x^2 dx

-2

value of integral = ?

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  • 9 years ago
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    y = √(4 - x²) for -2 ≤ x ≤ 2 is the top half of a circle of radius 2 centered at the origin. Using area, the integral is the area of a half circle of radius 2. This is

    2

    ∫ √(4 - x²) dx = ½ π(2)² = 2π.

    -2

    It might be more obvious that you're working with a circle if you do the following:

    y = √(4 - x²) ==> y² = 4 - x² ==> x² + y² = 4.

    That is the equation of the circle of radius 2 centered at (0,0). Since y is the positive root, you have the upper half. Since x varies from -2 to +2, you get the entire upper half.

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