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4.3 Riemann Sums and Definite Integrals
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Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a definite integral using properties of definite integrals.
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Riemann Sums 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. The entire interval is called the partition. Subintervals do not all have to be the same size.
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Riemann Sums Riemann Sums: Add areas of rectangles to estimate area. Rectangle widths don’t have to be the same. 3 basic types: –Left (use f(left endpoint) as height) –Right (use f(right endpoint) as height) –Midpoint (use f(midpoint) as height)
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Norm of the Partitional subinterval partition 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.
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Definition of a Definite Integral If f is defined on [a,b] and the limit exists then f is integrable on [a,b] and the limit is denoted by Longest rectangle width 0 # rectangles ∞ discretecontinuous
![Definition of a Definite Integral If f is defined on [a,b] and the limit exists then f is integrable on [a,b] and the limit is denoted by Longest rectangle width 0 # rectangles ∞ discretecontinuous](http://images.slideplayer.com/32/9805927/slides/slide_6.jpg "Definition of a Definite Integral If f is defined on [a,b] and the limit exists then f is integrable on [a,b] and the limit is denoted by Longest rectangle width 0 # rectangles ∞ discretecontinuous")
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Definite Integral Notation Leibniz introduced the simpler notation for the definite integral: Note that the very small change in x becomes dx.
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Theorem 4.4: Continuity Implies Integrability If a function f is continuous of [a,b], then f is integrable on [a,b].
![Theorem 4.4: Continuity Implies Integrability If a function f is continuous of [a,b], then f is integrable on [a,b].](http://images.slideplayer.com/32/9805927/slides/slide_8.jpg "Theorem 4.4: Continuity Implies Integrability If a function f is continuous of [a,b], then f is integrable on [a,b].")
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Example Evaluate the definite integral Remember: Why is it negative?
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Theorem 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 the vertical lines x=a and x=b is given by
![Theorem 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 the vertical lines x=a and x=b is given by](http://images.slideplayer.com/32/9805927/slides/slide_10.jpg "Theorem 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 the vertical lines x=a and x=b is given by")
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Example Consider the region bounded by the graph of f(x)=4x-x 2 and the x-axis.
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Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.
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Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.
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Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.
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Properties of Definite Integrals
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More Properties
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Homework 4.3 (page 272) #13-47 odd
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