Intuitive Integral
Evaluate the following integral to 4 decimal places.
∫ − 1 1 1 − x 2 d x
The answer is 1.5707.
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1 solution
This is a nice problem and a good reminder. Thanks for sharing it!
More generally, ∫ − a a a 2 − x 2 d x for a ≥ 0 is equivalent to the area of the semi-circle of radius a .
This area can then be written as a 2 × 2 π
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Thank you! I did not know that!
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This property actually follows from the circle equation: x 2 + y 2 = a 2 (a is the radius of the circle centered at the origin).
Rearranging the terms, we get y 2 = a 2 − x 2 . We get y = a 2 − x 2 (above the x-axis) or y = − a 2 − x 2 (below it).
From this, we can easily evaluate the integral.
1 − x 2 is the graph of a semi circle. The integral is asking for the area under the it, so it's just 2 π . I thought this would be a cool problem because people often forget what the integral is really evaluating. They are used to the brute force calculations, and don't take the time to think about what it's really asking.