MAT 21B Lecture Notes - Lecture 18: Moment Of Inertia, Weighted Arithmetic Mean, Space Elevator
MAT 21B – Lecture 18 – Application of Definite Integrals in Averages
• List of Applications of Definite Integrals
o 1) Quantities from rates over intervals
o 2) Area of any shape
o 3) Volume of any solid
o 4) Length of any curve
o 5) Surface area of surfaces of revolution
o 6) Work
o 7) Moments (first moment: torque, second moment: rotational inertia)
o 8) Averages
• The average of y1,…yN is
. The weighted average of y1,…yN with weights
is
.
• The average value of the function f on [a, b] is approximately the average of the
value of f at N equally spaced points,
. The exact average is the limit as of these sums is,
or in shorthand,
• Example: The average of over [0, 1] is
• Example 2: The average of over [0, 1] is
• Notation:
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Mat 21b lecture 18 application of definite integrals in averages: list of applications of definite integrals. =(cid:3117) value of f at n equally spaced points, (cid:1858)(cid:4666)(cid:1853)+(cid:3029) (cid:3028) (cid:4667)(cid:3029) (cid:3028) The exact average is the limit as of these sums is, =(cid:2869) lim or in shorthand, (cid:2869)(cid:3029) (cid:3028) (cid:1858)(cid:4666)(cid:1876)(cid:4667)(cid:1876) (cid:1858)(cid:4666)(cid:1853)+(cid:3029) (cid:3028) (cid:4667)(cid:2869) (cid:3029)(cid:3028) =(cid:2869: example: the average of (cid:1876) over [0, 1] is (cid:2869)(cid:2869) (cid:2868) (cid:1876) (cid:1876) [(cid:2870)(cid:2871)(cid:1876)(cid:2871)/(cid:2870)](cid:2868)(cid:2869)= (cid:2870)(cid:2871)(cid:883)(cid:2871)/(cid:2870)= (cid:2870)(cid:2871): example 2: the average of (cid:1876)(cid:2870) over [0, 1] is (cid:2869)(cid:2869) (cid:2868) (cid:1876)(cid:2870) (cid:1876) = [(cid:2869)(cid:2871)(cid:1876)(cid:2871)](cid:2868)(cid:2869)= (cid:2869)(cid:2871)(cid:883)(cid:2871)= (cid:2869)(cid:2871). (cid:2869)(cid:2868: notation: = (cid:2869)(cid:3029) (cid:3028) (cid:1858)(cid:4666)(cid:1876)(cid:4667)(cid:1876) Volume (or area or length: example: calculate the center of mass, center of gravity, and centroid of a hypothetical space elevator shown in the following diagram below. The linear mass density, (cid:4666)(cid:4667)= (cid:2869) . (cid:2871)6 6 = (cid:3117)(cid:3118)(cid:4666)(cid:2871)6 6(cid:4667)(cid:4666)(cid:2871)6+6(cid:4667) [](cid:3122)(cid:3119)(cid:3122) =(cid:3117)(cid:3118)(cid:4666)(cid:2871)6(cid:3118) 6(cid:3118)(cid:4667: centroid = (cid:3049) (cid:3049)= (cid:3119)(cid:3122)(cid:3122) (cid:2871)6 6 (cid:3119)(cid:3122)(cid:3122) (cid:2869)(cid:2870)(cid:4666)(cid:885)6+6(cid:4667)=(cid:883)(cid:890)+(cid:885)=(cid:884)(cid:883).
