Presentation on theme: "Definite Integrals Sec. 5.2. When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. subinterval partition The width."— Presentation transcript:
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.
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 (dummy variable) It is called a dummy variable because the answer does not depend on the variable chosen.
We have the notation for integration, but we still need to learn how to evaluate the integral.
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.
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
Example: Find the area between the x-axis and the curve From to. Check the answer on the calculator! pos. neg.
Be careful! When asked to EVALUATE AN INTEGRAL, you are finding the area between the function and the x- axis. Areas ABOVE the x-axis are positive. Areas BELOW the x-axis are negative. When asked to find the AREA of a REGION, be sure to make all negative areas positive so you’ll get the true area!