1. 101 meters 0 3 9− 𝑥 2 −1 2 2𝑥+1 Concepts to know:
Estimating Integrals using Riemann sums (LRAM, RRAM, MRAM, and Trapezoidal approximations.)
Estimate using 4 midpoint rectangles.
101 meters
Using geometric formulas to calculate definite integrals. Graph functions and calculate their area.
− 𝑥 2
−1 2 2𝑥+1

2. −2 𝑒 𝑥 3 − (-2e-2) -14 14 a) v(t) = 3 𝑡 2 +6t−9 b) t>1
3. Calculating definite integrals using the fundamental theorem of calculus.
−4 5 𝑓(𝑥)𝑑𝑥 =−2
1 5 𝑓(𝑥)𝑑𝑥 =12
𝑑 𝑑𝑡
−2 𝑒 𝑥 3 − (-2e-2) If
and
-14
−4 1 𝑓(𝑥)𝑑𝑥 =______
Then…
𝑑 𝑑𝑡
1 −4 𝑓(𝑥)𝑑𝑥 =______ 14
And…
𝑑 𝑑𝑡
4. Identifying an integral as a limit of a Riemann sum.
lim 𝑛→∞ 𝑘=1 𝑛
−2 3 𝑥 2 𝑑𝑥
lim 𝑛→∞ 𝑘=1 𝑛
1 2 ( 2𝑥+1 )𝑑𝑥
5. Motion problems using the definite integral (either by solving for C or using the fundamental theorem.)
a) v(t) = 3 𝑡 t−9
b) t>1
c) x(t) = 𝑡 𝑡 −9t−27

3. = = = = = = = = = = = = = 1 6 𝑠𝑖𝑛3𝑥 2 | 1 6 (sin 3𝜋 2 )2 - 1 6 (sin0)2
7. Finding anti-derivatives using the reverse-chain rule (with or without u-substitution) – must know trigonometric derivatives/integrals, as well as ‘e’ and ln.
𝜋 2
1 6 𝑠𝑖𝑛3𝑥 2 | = =
1 6 (sin 3𝜋 2 ) (sin0)2 =
1 6 (−1) (0)2 =
1 6
ln⁡( 𝑥 3 −1) | 2 = =
𝑙𝑛 2 3 −1 −ln⁡( 0 3 −1) =
𝑙𝑛7 −ln⁡|−1|
𝜋 4
−1 2 𝑠𝑖𝑛2𝑥 −1 |
−1 2 𝑠𝑖𝑛2 𝜋 −1 −[ −1 2 𝑠𝑖𝑛2 𝜋 −1 ] = =
𝜋 6
− −1 −[ − −1 ] =
−1 2 − −1 3 − − = = =