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. 0 3 9− 𝑥 2 −1 2 2𝑥+1

−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 2 +6t−9 b) t>1 c) x(t) = 𝑡 3 +3 𝑡 2 −9t−27

= = = = = = = = = = = = = 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 )2 - 1 6 (sin0)2 = 1 6 (−1)2 - 1 6 (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 4 −1 −[ −1 2 𝑠𝑖𝑛2 𝜋 6 −1 ] = = 𝜋 6 −1 2 1 −1 −[ −1 2 3 2 −1 ] = −1 2 − −1 3 −1 2 + 3 3 −3+2 3 6 = = =