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Indefinite Integrals and Net Change The integral of a rate of change is the net change. Indefinite Integrals and Net Change The integral of a rate of change is the net change. 5.5 Review of Chain rule Follow the link to the slide. Then click on the figure to play the animation. A Figure 5.1.8 Figure 5.1.9 Section 5.1 · Figures 8, 9 Upper and lower estimates of the area in Example 2 A A Section 1 / Figure 1 Section 5.1 · Figure 12 Approximations of the area for n 2, 4, 8 and 12 rectangles Section 5.1 · Figure 13 Approximating rectangles when sample points are not endpoints 5. 1 Sigma notation The graph of a typical function y ? x over [a, b]. Partition [a, b] into n subintervals a x1 x2 …xn b. Select a number in each subinterval ck. Form the product f ck ?xk. Then take the sum of these products. 5.2 The curve of with rectangles from finer partitions of [a, b]. Finer partitions create more rectangles with shorter bases. The definite integral of f x on [a, b] If f x is non-negative, then the definite integral represents the area of the region under the curve and above the x-axis between the vertical lines x a and x b Rules for definite integrals 5.3 The Fundamental Theorem of Calculus Where f x is continuous on [a,b] and differentiable on a,b Find the derivative of the function: The Fundamental Theorem of Calculus Indefinite Integrals Definite Integrals Try These Answers If the function is non-negative, net area area. If the function has negative values, the integral must be split into separate parts determined by f x 0. Integrate one part where f x 0 and the other where f x 0. but If the function is non-negative, displacement distance. If the function has negative values, the integral Must be split into separate parts determined by v 0. Integrate one part where v 0 and the other where v 0. If F is the antiderivative of f Let u inside function of more complicated factor. Check by differentiation