Create Presentation
Download Presentation

Download Presentation
## 5.2 Definite Integrals

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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