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β€’ Know how to use the laws of integration to find anti-derivatives o βˆ«π‘›π‘›π‘‘π‘‘π‘‘π‘‘ = 𝑛𝑛𝑑𝑑 + 𝑐𝑐, 𝑛𝑛 βˆͺ𝑐𝑐 ∈ ℝ o βˆ«π‘‘π‘‘π‘›π‘›π‘‘π‘‘π‘‘π‘‘ =π‘₯π‘₯𝑛𝑛+1𝑛𝑛+1+ 𝑐𝑐, 𝑛𝑛 β‰  βˆ’1, 𝑐𝑐 ∈ ℝ o βˆ«π‘‘π‘‘βˆ’1𝑑𝑑𝑑𝑑 =∫1π‘₯π‘₯𝑑𝑑𝑑𝑑 = ln|𝑑𝑑|+ 𝑐𝑐, 𝑐𝑐 ∈ ℝ o ∫sin 𝑑𝑑𝑑𝑑𝑑𝑑 = βˆ’cos 𝑑𝑑 + 𝑐𝑐; 𝑐𝑐 ∈ ℝ o ∫cos 𝑑𝑑𝑑𝑑𝑑𝑑 = sin 𝑑𝑑 + 𝑐𝑐; 𝑐𝑐 ∈ ℝ β€’ Know how to find the parent function of an anti-derivative given information about the parent function o If 𝑓𝑓′′(𝑑𝑑)= 2, 𝑓𝑓′(2)= 5, and 𝑓𝑓(2)= 10, then find 𝑓𝑓(𝑑𝑑) β€’ Riemann Sum Over and Under Approximations o Left hand Riemann sums are under approximates if 𝑓𝑓(𝑑𝑑) is increasing o Left hand Riemann sums are over approximates if 𝑓𝑓(𝑑𝑑)is decreasing o Right hand Riemann sums are under approximates if 𝑓𝑓(𝑑𝑑)is decreasing o Right hand Riemann sums are over approximates if 𝑓𝑓(𝑑𝑑)is increasing o Midpoint Riemann sums are under approximates if 𝑓𝑓′′(𝑑𝑑)< 0 o Midpoint Riemann sums are over approximates if 𝑓𝑓′′(𝑑𝑑)> 0 o Trapezoidal Riemann sums are under approximates if 𝑓𝑓(𝑑𝑑) is concave down o Trapezoidal Riemann sums are over approximates if 𝑓𝑓(𝑑𝑑) is concave up β€’ Know the, and how to apply, the First Fundamental Theorem of Calculus o If 𝑓𝑓(𝑑𝑑) is continuous on an interval, [π‘Žπ‘Ž, 𝑏𝑏] and 𝐹𝐹(𝑑𝑑) is the anti-derivative of 𝑓𝑓(𝑑𝑑), then βˆ«π‘“π‘“(𝑑𝑑) 𝑑𝑑𝑑𝑑 = 𝐹𝐹(𝑏𝑏)βˆ’ 𝐹𝐹(π‘Žπ‘Ž)π‘π‘π‘Žπ‘Ž β€’ Know the properties of indefinite integrals, and how to apply them to solve problems when given limited information o βˆ«π‘“π‘“(𝑑𝑑) 𝑑𝑑𝑑𝑑 = 𝐹𝐹(π‘Žπ‘Ž)βˆ’ 𝐹𝐹(π‘Žπ‘Ž)= 0π‘Žπ‘Žπ‘Žπ‘Ž o Given that π‘Žπ‘Ž < 𝑐𝑐 < 𝑏𝑏,βˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ =βˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ +βˆ«π‘“π‘“(𝑑𝑑)𝑏𝑏𝑐𝑐𝑑𝑑𝑑𝑑 o If, βˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ = 𝐾𝐾, then βˆ«π‘“π‘“(𝑑𝑑)π‘Žπ‘Žπ‘π‘π‘‘π‘‘π‘‘π‘‘ = βˆ’πΎπΎ o Given that 𝑏𝑏 < π‘Žπ‘Ž, then βˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ = βˆ’βˆ«π‘“π‘“(𝑑𝑑)π‘Žπ‘Žπ‘π‘π‘‘π‘‘π‘‘π‘‘ o If 𝐾𝐾 is a constant, then βˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ = πΎπΎβˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ o ∫[𝑓𝑓(𝑑𝑑)Β± 𝑔𝑔(𝑑𝑑)]𝑑𝑑𝑑𝑑 =βˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ Β±βˆ«π‘”π‘”(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘π‘π‘π‘Žπ‘Ž o Given that 𝑓𝑓(𝑑𝑑) is an even function, βˆ«π‘“π‘“(𝑑𝑑)π‘Žπ‘Žβˆ’π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ = 2βˆ«π‘“π‘“(𝑑𝑑)π‘Žπ‘Ž0𝑑𝑑𝑑𝑑 = 2βˆ«π‘“π‘“(𝑑𝑑)0βˆ’π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ o Given that 𝑓𝑓(𝑑𝑑) is an odd function, βˆ«π‘“π‘“(𝑑𝑑) 𝑑𝑑𝑑𝑑 = 0π‘Žπ‘Žβˆ’π‘Žπ‘Ž β€’ Know the formulas for displacement, net distance, and total distance o π·π·π·π·π·π·π·π·π·π·π‘Žπ‘Žπ‘π‘π·π·π·π·π·π·π‘›π‘›π·π· =βˆ«π‘£π‘£(𝐷𝐷)𝑑𝑑𝐷𝐷 π‘π‘π‘Žπ‘Ž, where 𝑣𝑣(𝐷𝐷) is a velocity function of time o 𝑁𝑁𝐷𝐷𝐷𝐷 π·π·π·π·π·π·π·π·π‘Žπ‘Žπ‘›π‘›π‘π‘π·π· = οΏ½βˆ«π‘£π‘£(𝐷𝐷)π‘‘π‘‘π·π·π‘π‘π‘Žπ‘ŽοΏ½, where 𝑣𝑣(𝐷𝐷) is a velocity function of time o π‘‡π‘‡π‘œπ‘œπ·π·π‘Žπ‘Žπ·π· π·π·π·π·π·π·π·π·π‘Žπ‘Žπ‘›π‘›π‘π‘π·π· =∫|𝑣𝑣(𝐷𝐷)|π‘‘π‘‘π·π·π‘π‘π‘Žπ‘Ž, where 𝑣𝑣(𝐷𝐷) is a velocity function of timeβ€’ Know the relationship between integrals and derivatives, and know how to apply them to word problems o The integral is the accumulation of change, the derivative is a rate of change o The derivative of the integral is the parent function  𝑑𝑑𝑑𝑑π‘₯π‘₯[βˆ«π‘“π‘“(𝑑𝑑)𝑑𝑑𝑑𝑑]= 𝑓𝑓(𝑑𝑑) o The integral of the derivative is the parent function  βˆ«π‘‘π‘‘π‘‘π‘‘π‘₯π‘₯[𝑓𝑓(𝑑𝑑)] 𝑑𝑑𝑑𝑑 = 𝑓𝑓(𝑑𝑑) β€’ Know how to find the average value of a function o 1π‘π‘βˆ’π‘Žπ‘Žβˆ«π‘“π‘“(𝑑𝑑)π‘π‘π‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘ = π·π·β„Žπ·π· π‘Žπ‘Žπ‘£π‘£π·π·π‘Žπ‘Žπ‘Žπ‘Žπ‘”π‘”π·π· π‘£π‘£π‘Žπ‘Žπ·π·π‘£π‘£π·π· π‘œπ‘œπ‘“π‘“ π‘Žπ‘Ž π‘“π‘“π‘£π‘£π‘›π‘›π‘π‘π·π·π·π·π‘œπ‘œπ‘›π‘› π‘œπ‘œπ‘›π‘› π‘Žπ‘Žπ‘›π‘›

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