Did more examples from the AP test involving area between 2 curves. Then learned about accumulation functions, which are a new class of functions for us. The input for an accumulation function is the 'b' value of a definite integral, which calculates the area under some other function. Thus the value of the accumulation function varies as the area under the curve varies. We explored an accumulation function that was calculating the area under a linear piecewise function, so we could use geometry to find the required values. This leads to the development of the fundamental theorem of calculus, which we started to develop using the difference between close input values of an accumulation function.
Notes from board
Blank copy of handout we started
Homework
#7-12 on the area handout (blank copy) (number answers)
Resources
accumulation functions introduction: link
FTC proof at your own pace: link
help with area;
simple area under a curve: link
area between two curves, unified region: link link 2
area between two curves: split region, trig: link
Learned how to find area using horizontally oriented rectangles. The paradigm is to integrate x (horizontal distance) expressed as a function of y, times dy (the height of the rectangles). For area between curves, it is "right minus left" to express the rectangle width.
Notes from board
video similar to topic
Continued proof of the FTC from Tuesday, including how it, along with properties of definite integrals, proves the FTC2 that we use to evaluate definite integrals. Used the FTC to find derivatives of accumulation functions (basically replace the t-variable with the x-variable upper bound). Sometimes the chain rule is involved, and sometimes both bounds are variables and have to be split into 2 integrals with an arbitrary "pit stop" number in between. Used the FTC and the hierarchy of functions to investigate a graph under which an accumulation function was defined and this allowed us to describe the accumulation function's behavior via attributes of the secondary function. Passed out practice test.
Notes from board (includes optional practice probs)
Homework
do the practice test, try optional practice problems (blank copy) (SOLUTIONS)
test is Tuesday
Resources
I-A4a: area under the curve using definite integration: link
I-A4b: area between two curves: link link2
I-U4: FTC, algebraically: link1 link2
I-U7: properties of definite integrals: link1 link2 (starts at 4:04)
I-U5: evaluating definite integrals: link1 link2 link3
I-A3: finding C: link
I-U3b: midpoint approximation: link
I-U3c: trapezoid from table: link