Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.9 Numerical Integration - Exercises - Page 457: 28


The average temperature is approximately $21.21^{\circ}$ Celsius.

Work Step by Step

Simpson's Rule states that $T_{n}=\dfrac{1}{3}[y_0+4y_1+2y_2+..+4y_{N-3}+2y_{N-2}+4y_{N-1}+y_N]\Delta x$ Since, $\Delta t=15 \ min=\dfrac{1}{4} \ hour $ Thus, using Simpson's Rule, we have: $S_{12}= \dfrac{1}{3}[v_0+4v_1+2v_2+..+4v_{9}+2v_{10}+4v_{11}+v_{12}]\Delta t$ Now, $T_{avg}=\dfrac{1}{3} S_{12}=(\dfrac{1}{3}) (\dfrac{1}{3})(\dfrac{1}{4} )[21+4(21.3)+2(21.5)+4(21.8)+2(21.6)+4(21.2)+2(20.8)+4(20.6)+2(20.9) +4(21.2)+2(21.1)+4(21.3)+21.2] \approx 21.21$ Hence, the average temperature is approximately $21.21^{\circ}$ Celsius.
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