ISHIK UNIVERSITY FACULTY OF EDUCATION Mathematics Education Department Orhan TUĞ (PhDc) Integratıon, fınıte sum and defınıte ıntegral 1 1

Upper and lower estimates of the area Figure 5.1.9 Figure 5.1.8 Error analysis Error analysis

Upper and lower estimates of the area in Example 2

Approximations of the area for n = 2, 4, 8 and 12 rectangles Section 1 / Figure 1

Approximating rectangles when sample points are not endpoints

5. 1 Sigma notation

Some important sum of sequences

The graph of a typical function y = ƒ(x) over [a, b] The graph of a typical function y = ƒ(x) over [a, b]. (P)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.

The curve of with rectangles from suitable partitions of [a, b] The curve of with rectangles from suitable partitions of [a, b]. Suitable partitions create more rectangles with shorter bases.

Definite integration ***The definite integral of f(x) with respect to x

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 If f(x) is continious on an interval [a,b], then its definite integral obver [a,b] exists.

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. but 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.

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

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

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]