Follow the link to the slide. Then click on the figure to play the animation. A Figure 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 = ƒ(x) over [a, b]. Partition [a, b] into n subintervals a < x 1 < x 2 <…x n < b. Select a number in each subinterval c k. Form the product f(c k )  x k. 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

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Answers

Indefinite Integrals and Net Change The integral of a rate of change is the net change. 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

Indefinite Integrals and Net Change The integral of a rate of change is the net change. 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.

5.5 Review of Chain rule

If F is the antiderivative of f

Let u = inside function of more complicated factor. Check by differentiation

Let u = inside function of more complicated factor. Check by differentiation

Substitution with definite integrals Using a change in limits

The average value of a function on [a, b]