AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
We can now write that after 1 second, distance traveled = 80−32t dt = 64 feet.
∫
1
0
( )
Where is the rock after 2 seconds? We use the calculator with 100 subintervals to
compute the average of RHS and LHS. Using our definite integral notation, we obtain
2
( 80−32t)
dt = 96 feet. (Again, we confirm geometrically that the area under the
∫
0
graph, shown in Figure 2, is 96.)
Figure 2: The graph of v=80–32t for 0 ≤ t ≤ 2
After 2 seconds, distance traveled = ∫ 80−32t dt 96 feet.
2
0
( ) =
The next question is to find out where the rock is after 3 seconds. So we again change
the upper limit of integration and use the calculator to compute Riemann sums to
3
3
approximate ∫ ( 80−32t)
dt (see Figure 3). We get ∫ ( 80−32t) dt = 96 feet.
0
0
Figure 3: The graph of v = 80−32 t for 0 ≤ t ≤ 3
This is silly; the rock has obviously moved during the third second. How can
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AP® Calculus: 2006–2007 Workshop Materials