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5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington

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When we find the area under a curve by adding rectangles, the answer is called a Rieman 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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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. if P is a partition of the interval

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is called the definite integral of over. If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by:

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Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

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Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen.

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We have the notation for integration, but we still need to learn how to evaluate the integral.

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time velocity After 4 seconds, the object has gone 12 feet. In section 5.1, we considered an object moving at a constant rate of 3 ft/sec. Since rate. time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

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If the velocity varies: Distance: ( C=0 since s=0 at t=0 ) After 4 seconds: The distance is still equal to the area under the curve! Notice that the area is a trapezoid.

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What if: We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example. It seems reasonable that the distance will equal the area under the curve.

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The area under the curve We can use anti-derivatives to find the area under a curve!

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Lets look at it another way: Let area under the curve from a to x. ( a is a constant) Then:

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min f max f The area of a rectangle drawn under the curve would be less than the actual area under the curve. The area of a rectangle drawn above the curve would be more than the actual area under the curve. h

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As h gets smaller, min f and max f get closer together. This is the definition of derivative! Take the anti-derivative of both sides to find an explicit formula for area. initial value

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As h gets smaller, min f and max f get closer together. Area under curve from a to x = antiderivative at x minus antiderivative at a.

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Area

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Area from x=0 to x=1 Example: Find the area under the curve from x = 1 to x = 2. Area from x=0 to x=2 Area under the curve from x = 1 to x = 2.

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Example: Find the area under the curve from x = 1 to x = 2. To do the same problem on the TI-89: ENTER 7 2nd

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Example: Find the area between the x-axis and the curve from to. On the TI-89: If you use the absolute value function, you dont need to find the roots. pos. neg.

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