5.2 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 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.
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. subinterval partition 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.
It is called a dummy variable because the answer does not depend on the variable chosen. upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration
We have the notation for integration, but we still need to learn how to evaluate the integral.
velocity time 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. After 4 seconds, the object has gone 12 feet.
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.
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.
The area under the curve We can use anti-derivatives to find the area under a curve!
Let area under the curve from a to x. (“a” is a constant) Let’s look at it another way: Then:
min f max f h The area of a rectangle drawn under the curve would be less than the actual area under the curve. The area of a rectangle drawn above the curve would be more than the actual area under the curve.
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.
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.
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
ENTER 2nd 7 Example: Find the area under the curve from x=1 to x=2. To do the same problem on the TI-89:
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