Antiderivatives from Slope and the Indefinite Integral

We have seen that the definite integral of a derivative gets us back to the original function. In other words, if f ' is the derivative of f, then f is an antiderivative of f '. For example:

if f (x) = x²   then  f '(x) = 2x
So an antiderivative of 2x is x². But, since the derivative of a constant is zero, x² + 1 is also an antiderivative of 2x, and so is x² + C, where C is a constant. Hence the general antiderivative of a function is a family of functions, which all differ by a constant.

This page explores finding the antiderivative graphically, thinking of the integrand as representing the slope of the antiderivative. The notation for the general antiderivative of a function, g, is which is called the indefinite integral because there are no limits. Using our example from above, we would write

This device cannot display Java animations. The above is a substitute static image
 See About the calculus applets for operating instructions.

## 1. Constant function

The applet shows a graph on the left of the integrand f ' (x) = 2, a constant function. On the right is the graph of the antiderivative: Think of the graph on the left (the integrand) as representing the slope of the graph on the right (the antiderivative). Note that since the left-hand graph is constant, so is the slope of the right hand graph, and we get a line with slope 2. Move the C slider; what happens to the graph? If you work backwards, thinking that the graph on the left is the derivative of the graph on the right, you see why changing C has no effect on the left-hand graph.

## 2. Different slopes

Select the second example from the drop down menu. Now the integrand changes value from -1 to 1 at x = 0. The antiderivative on the right therefore changes slope from -1 to 1 at x = 0. Move the x slider, causing a crosshair to move on the left-hand graph (it's hard to see on this example) and a point and a black tangent line to move on the right-hand graph. You can see from the tangent line that the slope of the antiderivative changes.

## 3. Changing slopes

Select the third example from the drop down menu. In this example, the integrand function changes from a constant value of 1 to a downward sloping line at x = 1. Move the x slider and watch what happens to the tangent line on the antiderivative graph. It starts out with a slope of 1, but then the tangent line's slope decreases to zero. Where does this happen on the left-hand graph? What happens if you move the x slider past this point? Think about the graph of the integrand as telling you the slope of the antiderivative at any given x value. The result is a family of antiderivatives, and the specific member of this family that is graphed depends on the value of C (move the C slider to change the graph to a different member of the family).

## 4. Changing slopes revisited

Select the fourth example, showing an integrand made up of two lines. Move the x slider to observe how the value of the integrand (on the left) tells you the slope of the antiderivative (on the right).

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## Acknowledgements

Derived from the work of Thomas S. Downey under a Creative Commons Attribution 3.0 License.