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Chapter 5 Integral

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Estimating with Finite Sums

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Approach

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Approach (2)

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Both approach are called Upper sum because they are obtained by taking the height of each rectangle as the maximum (uppermost) value of ƒ(x) for x a point in the base interval of the rectangle. Now, we will be using what so called lower sum

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Therefore

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Midpoint approach

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Conclusions:

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Distance travelled Suppose we know the velocity function y(t) of a car moving down a highway, without changing direction, and want to know how far it traveled between times t=a and t=b If we already known an antiderivative F(t) of v(t) we can find the car’s position function s(t) by setting s(t)=F(t)+C. The travelled distance is s(b)-s(a) How to calculate in case we have no formula s(t)? We need an approach in calculating s(t)

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approach Subdivide the interval [a, b] into short time intervals on each of which the velocity is considered to be fairly constant. distance = velocity x time Total distance

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Average Value of a Nonnegative Function

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Sigma Notation and Limits of Finite Sums

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Limits of Finite Sums Solution: We start by subdividing [0, 1] into n equal width subintervals The lower sum of rectangular is :

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Riemann Sums

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Riemann Sums(2) the width of the kth subinterval is

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Riemann Sums(3)

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Riemann Sums (4) Among three figures, which one gives us the most accurate calculation?

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Riemann Sums (5) In previous calculation, we can improve accuracy by increasing number of interval (n). However, in Reimann sum, we can go to more accurate calculation by making |P| goes to zero We define the norm of a partition P, written |P| to be the largest of all the subinterval widths. If |P| is a small number, then all of the subintervals in the partition P have a small width.

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The Definite Integral

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Notation and existence of definite Integrals

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Properties of Definite Integrals

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Average Value of a Continuous Function Revisited

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The Fundamental Theorem of Calculus

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The Fundamental Theorem of Calculus (2) But remember this

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Indefinite Integrals and the Substitution Rule Symbol

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Substitution: Running the Chain Rule Backwards

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Definite Integrals of Symmetric Functions

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Areas Between Curves

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But, be careful with this circumstances

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Integration with Respect to y

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Example : previous problem, but integration respects to y

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Combining Integrals with Formulas from Geometry

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