The definite integral
The indefinite integral is easy, all you do is integrate for a function + C. We don't see these boys again until we hit u-substitution and differential equations. The definite integral is what does all the heavy work in calculus. The definite integral is written as (Notation below) such that a function is continuous for a≤x≤b.
RECALL
Properties
Know these.
Practice by solving these two original problems.
Graphing
Looking at these original examples, one can easily plug in x,y values to graph, shade the area bounded and then find the area.
We see the shapes beneath the graph are a series of trapezoids. Therefore, we can use the trapezoid formula to solve for the area beneath the curve.
|
The function tells us that from 0 to 5, the function is just 3. So we need only to graph a horizontal line between that area and shade down to the x-axis. Now we use the rectangular area formula to find the area beneath the line.
|
Left/Right Hand Sums Using Sigma Notation
The definite integral is equal to the left/right hand sums when n subdivision from a to be inclusively approaches infinity. These sums are called Riemann Sum. f is "integrand" and a and b are "limits of integration." These sums will give the exact area under a curve from x=a to x=b.
NOTE:
* The default notation for the definite integral remains the same when f(x) is positive and when a is less than b.
|
** When f(x) is positive some some x values but negative for other x values with a still less than b, then the definite integral is equal to the sum of the areas above the x-axis (+ counted) and below the x-axis (- counted).
|