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Announcements Topics: -sections 7.3 (definite integrals) and 7.4 (FTC) * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

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Area How do we calculate the area of some irregular shape? For example, how do we calculate the area under the graph of f on [a,b]?

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Area Approach: number of rectangles: width of each rectangle: We approximate the area using rectangles.

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Area Left-hand estimate: Let the height of each rectangle be given by the value of the function at the left endpoint of the interval.

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Area Left-hand estimate: Riemann Sum

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Area Right-hand estimate: Let the height of each rectangle be given by the value of the function at the right endpoint of the interval.

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Area Right-hand estimate: Riemann Sum

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Area Midpoint estimate: Let the height of each rectangle be given by the value of the function at the midpoint of the interval.

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Area Midpoint estimate: Riemann Sum

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Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

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Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

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Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact?

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Area How can we improve our estimation? Increase the number of rectangles!!! How do we make it exact? Let the number of rectangles go to infinity!!!

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Riemann Sums and the Definite Integral Definition: The definite integral of a function on the interval from a to b is defined as a limit of the Riemann sum where is some sample point in the interval and

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The Definite Integral Interpretation: If, then the definite integral is the area under the curve from a to b.

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Estimating a Definite Integral Estimate using left-endpoints, midpoints, and right-endpoints with n=4.

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The Definite Integral Interpretation: If is both positive and negative, then the definite integral represents the NET or SIGNED area, i.e. the area above the x-axis and below the graph of f minus the area below the x-axis and above f

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Evaluating Definite Integrals Example: Evaluate the following integrals by interpreting each in terms of area. (a)(b) (c)

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Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

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Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b.

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Summation Property of the Definite Integral (6) Suppose f(x) is continuous on the interval from a to b and that Then

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Properties of the Definite Integral (7) Suppose f(x) is continuous on the interval from a to b and that Then

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Types of Integrals Indefinite Integral Definite Integral antiderivative of f function of x number

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The Fundamental Theorem of Calculus If is continuous on then where is any antiderivative of, i.e.,

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Evaluating Definite Integrals Example: Evaluate each definite integral using the FTC. (a)(b) (c)(d)

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Evaluating Definite Integrals Example: Try to evaluate the following definite integral using the FTC. What is the problem?

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Differentiation and Integration as Inverse Processes If f is integrated and then differentiated, we arrive back at the original function f. If F is differentiated and then integrated, we arrive back at the original function F. FTC II FTC I

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The Definite Integral - Total Change Interpretation: The definite integral represents the total amount of change during some period of time. Total change in F between times a and b: value at end value at start rate of change

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Application – Total Change Example: Suppose that the growth rate of a fish is given by the differential equation where t is measured in years and L is measured in centimetres and the fish was 0.0 cm at age t=0 (time measured from fertilization).

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Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

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Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

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Application – Total Change (a) Determine the amount the fish grows between 2 and 5 years of age. (b) At approximately what age will the fish reach 45cm?

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