MAT 21B Lecture Notes - Lecture 16: Specific Weight, Summation, Standard Deviation
MAT 21B – Lecture 16 – Application of Definite Integrals in Moments
• Example: The first moment of the sequence x1 = 2, x2 = 5, x3 = 9 with weights
, respectively is: 2(7) + 5(3) + 9(1) = 38. This sum can
also be expressed with sigma notation such that
• The second moment from the example above is
• An application of first moments appears in the numerator of weighted averages.
Given the following table, the weighted average can be calculated.
Assignment
Score (%)
Weight on score (%)
Midterm #1
70
25
Quizzes
80
15
Let x1 = 70, , x2 = 80, and . Then the weighted average is
The numerator is the first moment.
• An application of the second moment appears in variance in standard deviation
• Problem: Another application of first moments appear in physics concept of
torque. Calculate the torque due to the wrench from following diagram:
The torque due to the person is The
wrench also has weight. Its linear weight density,
. The torque due to
the wrench, is where x is the length of the moment arm and
find more resources at oneclass.com
find more resources at oneclass.com
83
MAT 21B Full Course Notes
Verified Note
83 documents
Document Summary
Mat 21b lecture 16 application of definite integrals in moments: example: the first moment of the sequence x1 = 2, x2 = 5, x3 = 9 with weights. (cid:1875)(cid:2869)=(cid:889), (cid:1875)(cid:2870)=(cid:885), (cid:1875)(cid:2871)=(cid:883), respectively is: 2(7) + 5(3) + 9(1) = 38. Given the following table, the weighted average can be calculated. (cid:2871)(cid:2869) (cid:2871)(cid:2869) Let x1 = 70, (cid:1875)(cid:2869)=(cid:884)5, x2 = 80, and (cid:1875)(cid:2870)=(cid:883)5. The numerator (cid:1876)(cid:2869) (cid:1875)(cid:2869)+(cid:1876)(cid:2870) (cid:1875)(cid:2870) is the first moment. (cid:3051)(cid:3117) (cid:3050)(cid:3117)+(cid:3051)(cid:3118) (cid:3050)(cid:3118) (cid:2870)(cid:2873)+(cid:2869)(cid:2873: an application of the second moment appears in variance in standard deviation, problem: another application of first moments appear in physics concept of. Calculate the torque due to the wrench from following diagram: The torque due to the person is =(cid:883)5(cid:882) (cid:1864) (cid:883)(cid:882) (cid:1858)=(cid:883)5(cid:882)(cid:882) (cid:1858) (cid:1864). The torque due to the wrench, is = = (cid:1876) (cid:1875) where x is the length of the moment arm and dw is the weight of one rectangle (which was cut from vertical strips).
