Barrons AP Calculus - David Bock

(f) F(6) =

(g) In Figure N6–22b we evaluate the areas in the original graph.

FIGURE N6–22b

Measured from the lower limit of integration, 4, we have (with “f” as an abbreviation for “f (t)

dt”):

We note that, since F ′(= f ) is linear on (2,4), F is quadratic on (4,8); also, since F ′ is positive

and increasing on (2,3), the graph of F is increasing and concave up on (4,6), while since F ′ is

positive and decreasing on (3,4), the graph of F is increasing but concave down on (6,8).

Finally, since F ′ is constant on (4,6), F is linear on (8,12). (See Figure N6–22c.)

FIGURE N6–22c

Chapter Summary

In this chapter, we have reviewed definite integrals, starting with the Fundamental Theorem of
Calculus. We’ve looked at techniques for evaluating definite integrals algebraically, numerically, and
graphically. We’ve reviewed Riemann sums, including the left, right, and midpoint approximations as
well as the trapezoid rule. We have also looked at the average value of a function.
This chapter also reviewed integrals based on parametrically defined functions, a BC Calculus