Today: what an integral is, why F(b) - F(a) works, and Problem 1 line by line.
Let them watch the sums for N = 4, 8, 32 before the question.
Reveal: the sums climb to the area under the curve, e^-1 - e^-2.
Area below the axis counts negative. Then the three rules on the right.
Watch the shaded area grow and the graph of A trace out. Then predict.
Reveal: A'(t) = f(t); at t = 1.5 the slope is e^-1.5, about 0.2231.
Two antiderivatives, same rise between 1 and 2. That rise is the area.
Read each line. Predict the answer to the question before the reveal.
Reveal: a = 0 makes the integrand the constant e^-b, so the integral is e^-b.
The strips stretch by 2 and keep their heights. Say why dx = du/2.
Follow the interval from the x-axis to the u-axis. Predict before the reveal.
Reveal: no. The reversed limits and the minus sign together give the right number.
Five take-aways. Then go do the exercises.