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Warm Up

Warm Up. Using your calculator, find the integral of. 6 .5 Trapezoid Rule. Goal: Estimate the integral using the trapezoidal rule. . Why is this useful?. Using integrals to find area works extremely well as long as we can find the antiderivative of the function.

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Warm Up

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  1. Warm Up Using your calculator, find the integral of

  2. 6.5 Trapezoid Rule • Goal: Estimate the integral using the trapezoidal rule.

  3. Why is this useful? 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 don’t even have a function, but only measurements taken from a real-life object. What we need is an efficient method to estimate area when we can not find the antiderivative.

  4. We saw in the warm up…. • The actual area under the curve

  5. LRAM gives us: Approximate area: (too low)

  6. RRAM gives us: Approximate area: (too high)

  7. What if? Averaging the two gets us much closer to the exact area:

  8. 5.5 Trapezoid Rule xo=a x1 x2 x3 xn-1 xn = b

  9. Trapezoidal Rule: Where [a,b] is partitioned into n subintervals of equal length, and h = width of subinterval = (b – a)/n

  10. To see if the Trapezoidal Rule is an overestimate, underestimate, or exact, use the Concavity Test. If f’’(x) = 0, approximation is exact. If f’’(x) > 0, approximation is an overestimate If f’’(x) < 0, approximation is an underestimate.

  11. Example • Use the trapezoidal rule with n=4 to estimate Tehn, determine if it is an overestimate or underestimate.

  12. Solution • We need to split the interval [1,2] into 4 equal parts, then set up a table.

  13. We need to find h.

  14. Simply plug the numbers from the table into the formula.

  15. Continued The exact integral is 2.333. The approximation overestimates the integral by about half a percent.

  16. Check it • Look at Example 2 on page 312 to see how to use this rule given a table when there is no equation - It is even easier 

  17. Homework

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