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Definite Integrals Review (sections 5.6-5.10) AP Calculus Quiz Topics Fundamental Theorem of Calculus (section 5.6) Nderiv and FnInt on calculator Definite integral properties (section 5.7) Area using integrals (section 5.8) Example: Region bounded by y = -x^2 + 5 and y = -2x + 2 Volumes of solids (section 5.9) Fundamental Theorem of Calculus To evaluate the definite integral of f(x) = 𝑥 2 from -1 to 2: 2 2 𝑥 𝑑𝑥 −1 Let F(x) = 𝑥 2 𝑑𝑥 = Then (2)3 3 2 2 𝑥 𝑑𝑥 −1 − (−1)3 3 = 𝑥3 3 = F(2) – F(-1): 9 3 =3 Integral Properties If function values of f(x) are positive and the interval boundaries are increasing, the integral will be positive. If functional values of f(x) are negative (below x axis) and the interval boundaries are increasing, the integral will be negative. As a result, it is possible to have areas of positive and negative area “cancel,” resulting in an integral of 0. Odd/Even Integrals with Symmetric Limits Odd Function: f(-x) = -f(x) Even: f(-x) = f(x) ODD: EVEN: Examples: y = sin x, y = tan x Examples: y = cos x, 𝑦 = 𝑥 2 𝑦 = 𝑥3 ** Interval bounds must be symmetric!!! Reversal of limits of integration: 𝑏 𝑓 𝑎 𝑥 𝑑𝑥 = − 𝑎 𝑓(𝑥)dx 𝑏 Integral of Constant Times Function: 𝑏 𝑘 𝑎 ∙ 𝑓 𝑥 𝑑𝑥 = 𝑘 𝑏 𝑓(𝑥) 𝑎 dx Integral of Sum: 𝑏 𝑏 𝑓 𝑥 + 𝑔 𝑥 𝑑𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 + 𝑎 𝑔 𝑥 𝑑𝑥 𝑎 Sum of Integrals With Same Integrand 4 5 𝑓 𝑥 𝑑𝑥 + 1 5 𝑓 𝑥 𝑑𝑥 = 4 𝑓 𝑥 𝑑𝑥 1 (Also allows integral to be broken into more convenient parts) Use Common Sense! 2 | 0 𝑥 | dx 8 1 𝑑𝑥 0 Use Common Sense: Not all Integrals Involve Calculus! 2 | 0 𝑥 | dx 8 1 𝑑𝑥 0 Area Bounded By Curves Draw rectangular “strip” between curves and write formula for area using dx or dy for the width depending on the orientation. May need to find intersection points of functions to find interval boundaries. Area of “strip” = l x w = (y1 – y2)dx Write in terms of x: Integral is the SUM of strips over interval: 1 4 − 4𝑥 𝑑𝑥 = 2 (𝐴𝑟𝑒𝑎 𝑜𝑓 𝑟𝑒𝑔𝑖𝑜𝑛) 0 Area bounded by curves: “Sideways” Functions on left and right instead of top and bottom. Make sure “strip” always extends from one function to another (Not one function back to itself) 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑟𝑒𝑔𝑖𝑜𝑛: 𝑑 𝑓 𝑦 − 𝑔 𝑦 𝑑𝑦 𝑐 Integral Vs. Area Be able to write the INTEGRAL EXPRESSION for the area. Additional Topics Make sure you can find basic integrals (look over trig derivatives list – pay attention to SIGNS!) Chain Rule for integrals: (3𝑥 + 5)6 𝑑𝑥 “Tricky” integrals – change form, make graphs, etc: 𝑥 𝑑𝑥 1 𝑑𝑥 3 𝑥 𝑥 − 2 𝑥 + 1 𝑑𝑥 2𝑥(4𝑥 2 − 3)5 𝑑𝑥 1 𝑑𝑥 (2𝑥−5)2 2𝑥 − 1𝑑𝑥 Look out for odd/even functions (can graph to check) Draw original function given f’(x) Additional Topics Know how to find functional integral (fnInt) on graphing calculator Be able to write equations for simple volume of solids (disc method) Cylinder Volume: 𝜋 𝑟 2 ℎ Use trapezoidal rule/Riemann sums to estimate area using a table of values (no function given). (PLOT GRAPH!!!!) Application problems: Area under curve for velocity-time graph is displacement! Review Problems: pg. 261: R6 bcd, R7, R8, R10 (trap rule on part c)