5.6 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington.

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

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.

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

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:

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

Integration Symbol lower limit of integration upper limit of integration integrand variable of integration

Example: Express the following limit as a definite integral

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

Definitions So the area below the x-axis is considered negative… So evaluating the integral in reverse order negates the area…

Ex. If the velocity is: The distance is equal to the area under the curve! Notice that the area is a trapezoid. Find the distance traveled.

Ex. Use formulas from geometry to find: The definite integral is equal to the area under the curve!

Ex. Use formulas from geometry to find: