Calculus II Notes 6.1 - M408L Integral Calculus Dr. Berg...

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M408L Integral Calculus Dr. Berg Page 1 5.5 6.1 Area Between Curves To approximate the area between two curves, we can use rectangles as we did before. Suppose that f ( x ) g ( x ) on the interval [ a , b ] . Then, if we divide the interval into n intervals of equal width Δ x = b a n and let x n * be a sample point from the i th interval. Then the area is approximately f ( x i * ) g ( x i * ) [ ] Δ x i = 1 n . Theorem If f ( x ) g ( x ) on the interval [ a , b ] , then the area bounded by the curves f ( x ) and g ( x ) over the interval [ a , b ] is A = f ( x ) g ( x ) [ ] dx a b . The more general formulation follows. Theorem The area between the curves y = f ( x ) and y = g ( x ) between x = a and x = b is A = f ( x ) g ( x ) dx a b In general, it is necessary to find the intersections of the two curves and find which function dominates on each interval determined by the intersections. Also, when the problem calls for finding the area bounded by two curves, any region with unbounded integral (unbounded area) would be omitted.
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This note was uploaded on 02/28/2010 for the course M 56495 taught by Professor Berg during the Spring '10 term at University of Texas at Austin.

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Calculus II Notes 6.1 - M408L Integral Calculus Dr. Berg...

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