Evaluate definite integrals analytically using the Fundamental Theorem of Calculus.
I can apply rules for finding the antiderivative of a function.
I can use the FTC to evaluate definite integrals by finding the difference of the antiderivative evaluated at the limits of integration.
Quick Lesson Plan
A figure skater’s velocity function serves as the context for our second investigation of the Fundamental Theorem of Calculus. Students will use analytic methods (writing a possible position function), numeric methods (using Math:9 on their calculator or other numeric integration device), and verbal interpretations to construct meaning from the FTC. (We will explore graphical relationships in tomorrow’s lesson on Topic 6.8.) Then, using initial condition information (position), students find a particular solution. Today’s work serves as a precursor to Topic 6.8 and the all-important “+C”!
Groups easily discovered that the integral value given by their calculator (Activity question 2) was equal to the value found by substitution into their position function (Activity question 4). At this point, teachers should cultivate the “Aha!” moment in the classroom. This is an amazing result and should be celebrated! This is the conclusion to write in the margin for Activity question 5, as well. Be sure to point out that we are now able to find exact areas under graphs without resorting to geometric shapes and formulas.
If time permits, relate the rearrangement of the FTC to the slope-intercept form of a line, y = mx + b: F(a) is the initial condition or initial value, b, and the integral becomes the accumulated changed, mx, where the integrand fills the role of m, the rate of change, with dx directing distance along the x-axis.
Many, many, many non-calculator FRQs require the use of geometric regions between the rate of change graph and the x-axis when investigating functions defined by integrals. The FTC opens many options for students to evaluate integrals, but they must keep their geometry skills sharp, too!
“Be sure to point out that we are now able to find exact areas under graphs without resorting to geometric shapes and formulas.” But don’t let students assume that they will never need to use geometric shapes and formulas.