Warm Up 1) 2) 3)Approximate the area under the curve for 0 < t < 40, given by the data, above using a lower Reimann sum with 4 equal subintervals. 4)Approximate.

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Presentation transcript:

Warm Up 1) 2) 3)Approximate the area under the curve for 0 < t < 40, given by the data, above using a lower Reimann sum with 4 equal subintervals. 4)Approximate the area under the curve for 0 < t < 40, given by the data, above using a midpoint Reimann sum with 4 equal subintervals.

Area Properties of Definite Integrals

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So, what if the graph does not allow for the use of simple geometric area formulas? How do I get the area of a rectangle? How do we get a better area using these lower sum rectangles? What would make the area perfect?

So, the area beneath the curve (to an axis) is the definite integral! Newton vs. Leibniz

Set up a definite integral that yields the area of the shaded region.

f(x) = ½ x 2 + 3

Set up a definite integral that yields the area of the shaded region.

f(x) = sin x Find the value of the definite integral that you created. Did you get what you expected?

Evaluate the integral. Find the area between the x-axis and f(x) on the interval [0,9]

Properties of Integrals

If f is an even function, then If f is an odd function, then

Evaluate the integral.