Calc05_5 - 5.5 Numerical Integration concave down concave...

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Unformatted text preview: 5.5 Numerical Integration concave down concave up concave down concave up concave up concave up concave down Using integrals to find area works extremely well as long as we can find the antiderivative of the function. Sometimes, the function is too complicated to find the antiderivative. At other times, we dont even have a function, but only measurements taken from real life. What we need is an efficient method to estimate area when we can not find the antiderivative. Approximation of Area Under a Curve A more accurate approximation of area under a curve can be found by finding the area of trapezoids (rather than LRAM, RRAM, MRAM). Area of a trapezoid = h 2 b 1 + b 2 ( 29 [ ] 2 Approximate the area under the curve y = x , on the interval 1,2 , by dividing the curve into 4 equal lengths and using the area of trapezoids. [ ] 2 Approximate the area under the curve y = x , on the interval 1,2 , by dividing the curve into 4 equal lengths and using the area of trapezoids. 2 2 2 2 2 2 1 5 6 7 Area = 1 + 2 + + + 4 8 4 4 4 1 110 = 5 + = 2.34375 8 8 Trapezoidal Rule: ( 29 1 2 1 2 2 ... 2 2 n n h T y y y y y- = + + + + + ( h = width of subinterval ) This gives us a better approximation than either left or right rectangles....
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Calc05_5 - 5.5 Numerical Integration concave down concave...

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