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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.**

The width of a rectangle is called a subinterval. The entire interval is called the partition. subinterval 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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**variable of integration**

upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration 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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**Since rate . time = distance:**

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. time velocity After 4 seconds, the object has gone 12 feet.

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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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**Let’s look at it another way:**

Let area under the curve from a to x. (“a” is a constant) Then:

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max f The area of a rectangle drawn under the curve would be less than the actual area under the curve. min f 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! initial value Take the anti-derivative of both sides to find an explicit formula for area.

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

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**Find the area under the curve from x=1 to x=2.**

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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**p Example: Find the area between the x-axis and the curve from to .**

pos. neg. On the TI-89: If you use the absolute value function, you don’t need to find the roots. p

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