Warm up was to practice deciphering/explaining the Riemann definition of definite integral. Separated the ideas of indefinite integrals from definite formally, then worked through right and midpoint rectangle approximations. Finally started looking at how to move fluently between the infinite sum representation to/from the definite integral representation algebraically.
Notes from board
Homework
p. 263 #25-30, 33, 34 (due Tuesday)
Resources
LRAM and RRAM example by me :) link
another RRAM example by me :) link
more good examples, by another person: link
Warm up was to find C involving position/velocity. Went over homework, then did some more translating between the Riemann infinite sum and the definite integral. Looked at integration through a numerical view with a table and emphasized keeping track of the delta-x values as they are often non-constant when in table form. Looked at yet another way to approximate integrals, which is to use trapezoids. Derived the trapezoid rule in the abstract, then worked 2 examples using it and also a third involving a table of values.
Notes from board
Homework
#1-12 on this handout (link) (numerical answers)
Resources
write a definite integral from an infinite Riemann sum: link1 link2
write an infinite Riemann sum from a definite integral: link
LRAM and RRAM from table: link (ignore the 1/10th in the 2nd example)
Midpoint sum given a function: link (first half)
trapezoid rule examples: link1 link2 (starts at 5:27)
Learned about when LRAM/RRAM/TRAP approximations of definite integrals are over/under estimates based on inc/dec or concave up/down. Did a midpoint estimate from a table, then thought about a couple of "riddle" problems which emphasized that a definite integral deals with area under a curve between two boundaries.
Looked at velocity and saw that the area under the velocity curve is the displacement. Used this idea along with integrating to go from velocity to position and subtracting two positions to arrive at the Fundamental Theorem of Calculus Part 2 which allows us to evaluate definite integrals to find exact area, not just estimates. It involves finding the antiderivative, plugging in the boundaries, and subtracting them. Worked a few examples, then closed with moving between infinite sum and definite integral representations.
Notes from board
Homework
p. 288 #5-20
test Monday on basic antiderivatives, rev chain rule, and u-sub
HW Resources
basis for second FTC if you're lost: link
basic definite integral evaluation; link
another definite integral example: link
Test Study Resources
rationale behind reverse power rule: link
many great integral examples link
reverse chain rule concept video example video
some more examples link link 2, cool old australian guy
U-Substitution example link
Some more: link1 (starts at 6:30) link2 (starts at 7:20, alt method)