Presentation on theme: "5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington."— Presentation transcript:
15.2 Definite IntegralsGreg Kelly, Hanford High School, Richland, Washington
2When 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.subintervalpartitionSubintervals do not all have to be the same size.
3subintervalpartitionIf the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted byAs gets smaller, the approximation for the area gets better.if P is a partitionof the interval
4is called the definite integral of overIf we use subintervals of equal length, then the length of a subinterval is:The definite integral is then given by:
5Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.
6variable of integration upper limit of integrationIntegrationSymbolintegrandvariable of integration(dummy variable)lower limit of integrationIt is called a dummy variable because the answer does not depend on the variable chosen.
7We have the notation for integration, but we still need to learn how to evaluate the integral.
8Since 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.timevelocityAfter 4 seconds, the object has gone 12 feet.
9If 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.
10What 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.
11The area under the curve We can use anti-derivatives to find the area under a curve!
12Let’s look at it another way: Let area under the curve from a to x.(“a” is a constant)Then:
13max fThe area of a rectangle drawn under the curve would be less than the actual area under the curve.min fThe area of a rectangle drawn above the curve would be more than the actual area under the curve.h
14As h gets smaller, min f and max f get closer together. This is the definition of derivative!initial valueTake the anti-derivative of both sides to find an explicit formula for area.
15As 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.