Chapter 5 AP Calculus BC

5.1 Estimating with Finite Sums Drive 2 hours at 75 mph…..Distance Traveled?? Graph…. Graph it for x from 0 to 5 5 subintervals…….. Example: LRAM RRAM MRAM Do some examples:

5.2 Definite Integrals Riemann Sums convert to Integrals…. Parts of an Integral: a is the lower limit b is the upper limit f(x) is the Integrand dx – variable of integration All continuous functions are integrable. Area under a curve – If y = f(x) is nonnegative and integrable over a closed interval [a,b] then the area under the curve y = f(x) from a to b is:

5.2 cont’d. When f(x)<0, then the integral will come out negative. Area: Area above – area below If f(x) = c, where c is a constant, on the interval [a,b] then: Calculator – fnint(f(x), x,a,b) Examples: Riemann Sums

5.3 Definite Integrals/Antiderivatives Properties: Order of Integration: Zero: Constant Multiple: Sum/Difference: Additivity: Find: Example: Given:

5.3 cont’d. Average Value Theorem- If f is integrable on [a,b] its average value on [a,b] is Find av(f) and does f ever take on this value in the interval? Mean Value Theorem for Definite Integrals – If f is cts. On [a,b] then at some point c in [a,b] Fundamental Theorem of Calculus More in 5.4

5.4 Fundamental Theorem of Calculus EXAMPLES: Part 2: Integral Evaluation Theorem Evaluate: Find the area of the region between And the x-axis on [0,3]

5.5 Trapezoidal Rule WHY? To Approximate: Use the Trapezoidal Rule… Examples: Use Trap. Rule with n=4……… Find the actual answers.