MAT 21B Lecture Notes - Lecture 19: Area Density
MAT 21B – Lecture 19 – Review of Applications of Definite Integrals
• The nine applications of definite integrals convey the two messages
o 1) Definite integrals arise in applications as the sum of infinitely many
infinitesimally small quantities
o 2) The logic and reasoning of setting up these sums is presented either
with the 1) Language of approximations, and 2) Language of differentials
• Recap of Application #1: Quantities from Rates
• Review Problem 1: The torque produced by a wrench
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MAT 21B Full Course Notes
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=(cid:2869: review problem 1: the torque produced by a wrench (cid:4666)(cid:1853)+(cid:3029) (cid:3028) (cid:4667)(cid:3029) (cid:3028) = (cid:3029)(cid:3028) (cid:1856)(cid:1872)= (cid:1856)(cid:3029)(cid:3028) In this case, (cid:1870)(cid:4666)(cid:1876)(cid:4667)=(cid:1876)+4 since the coordinate system shifted to the left by 4 units such that it began at x = This can be verified such that r(-4) = 0 and this makes sense since the length of the moment arm should be 0 at the first value of x. Similarly, if the coordinate system is set to begin at y = -16, then r(y) = y + 16: review problem 2: the surface area, a of a slanted edge on a dam. = (cid:1875)(cid:4666)(cid:1877)(cid:4667) (cid:1872)(cid:1853)(cid:2870)+(cid:883) (cid:1856)(cid:1877)= (cid:1875)(cid:4666)(cid:1877)(cid:4667)(cid:1871)(cid:1857)(cid:1855) (cid:1856)(cid:1877)=(cid:1871)(cid:1857)(cid:1855) (cid:1875)(cid:4666)(cid:1877)(cid:4667) (cid:1856)(cid:1877): review problem 3: use the language of differentials to set up a definite integral equal to the surface area of the surface obtained by revolving the graph of f(x) = 1 x2 on [0, 1] about the x-axis.
