Ch. 6 – The Definite Integral 6.5 – Trapezoidal Rule

Cut into 6ths, then make trapezoids using the L and R endpoints! Now that we learned how to find exact area under a curve using integrals, let’s estimate area with trapezoids? Boooo! Recall: area of a trapezoid = Ex: Estimate the area of the region bounded by the graph of f(x) = 9 – x2 and the x- and y-axes using the trapezoidal rule and 6 subintervals. x = 3   Cut into 6ths, then make trapezoids using the L and R endpoints!

Ex (cont’d): Note that this estimate is equal to f(x) 9 0.5 8.75 1 8 1.5 6.75 2 5 2.5 2.75 3 Ex (cont’d): Note that this estimate is equal to the average of LRAM6 & RRAM6! Is this estimate an overestimate or an underestimate? It’s an underestimate because the curve is concave down!   x = 3

Ex (cont’d): The actual area of f(x) = 9 – x2 over [0, 3] is 18 Ex (cont’d): The actual area of f(x) = 9 – x2 over [0, 3] is 18. Use that answer to find the percent error in the trapezoidal rule estimate found on the last slide. So the trapezoidal sum underestimated the area by 0.694%.

Ex: The temperature in my classroom from 6am to 6pm is shown below Ex: The temperature in my classroom from 6am to 6pm is shown below. Use the Trapezoidal Rule to approximate the average temperature for the 12 hour period. Average temperature = average value! Since we don’t have a function to integrate, let’s use trapezoid rule to approximate the area underneath the curve! Time (h after 6am) 1 2 3 4 5 6 7 8 9 10 11 12 Temp (°F) 54 60 71 70 68 65 64 63 66 67 69 72

Simpson’s Rule = To approximate Simpson’s Rule = To approximate , partition [a, b] into an even # (n) of subintervals of length h = (b-a)/n: Use the program on your calculator! Show what h and n are for your work! Ex: Use Simpson’s Rule to approximate the area underneath y = 9 – x2 using 6 subintervals over [0, 3]. n = 6, h = ½ …