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If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation for the area gets better. if P is a partition of the interval Subinterval (partition)

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Learning Target: The definite integral is limit of the Riemann Sum Integral measures accumulated change. 4.3 Definite Integrals

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When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. Subinterval (partition) The width of a rectangle is called a subinterval. Subintervals do not all have to be the same size.

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The Definite Integral Leibnitz introduced this simpler notation to represent the Rimeann Sum Note that the very small change in x becomes dx. The limit of the Riemann Sum, as Partition width goes to zeo

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Integration Symbol lower limit of integration upper limit of integration integrand variable of integration

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The Definite Integral as the Area of a Region: If f is continuous and nonnegative on [a, b], then the area of the region bounded by the graph of f, the x-axis and x = a and x = b is …

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1. Set up a definite integral that yields the area of the region. 2. Use a geometric formula to evaluate the integral.

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“Negative” area Regions below the x-axis have “negative” area. Why? p. 274 #47.

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Properties of Definite Integrals: 2. If the upper and lower limits are equal, then the integral is zero. 1. Reversing the limits changes the sign. 3. Constant multiples can be moved outside.

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1. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 3. Constant multiples can be moved outside. 4. Integrals can be added and subtracted.

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4. Integrals can be added and subtracted. 5. Intervals can be added (or subtracted.)

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Evaluate the integral using the following values:

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4.3 Definite Integrals If I have ever made any valuable discoveries, it has been owing more to patient attention, than to any other talent. Homework: p. 273 #17-23(odd), 29-43(odd) Homework: p. 273 #17-23(odd), 29-43(odd)

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