# Integrals

## Upper and Lower Sums

Here, we show an experimental introduction of the concept of definite and indefinite integrals using upper and lower sums (see Hohenwarter, Jauck, Lindner, 2005). In the dynamic construction below you see a map of a large piece of real estate that is bounded by a river on one of its sides. How do we calculate its area?

In the mathlet we used a parabola as a model for the river. You can adapt this parabola by dragging the points `P`_{1}, `P`_{2} and `P`_{3} with the mouse. Thus we transformed the question into: How do we calculate the area below the parabola? This could be the starting point to discuss different possibilities of approximating this area with your students. In the applet below you can use GeoGebra's lower and upper sums of a function by moving the slider `n`. In this way you can vary the number of rectangles used for the approximation of the area, what could be the basis to motivate and introduce the definite Riemann integral to your students.

Editor's note, May 2014: For an HTML5 version of the above applet,
click here.

## Antiderivative

Finally, you can also use GeoGebra to visualize the concept of an antiderivative. We assume that the students already know how to calculate the definite integral (e.g. by approximating it using upper and lower sums). This lets us use the idea of an *area function* `Area`(`b`) for the area between a given function `f` and the `x`-axis within a certain interval [`a`, `b`]. In the applet below we drew a point `Area` whose `y`-coordinate is the signed area within the interval between point `A` and point `B`. If you drag point `B` you get the graph of the area function as the trace of point `Area`. When your students try to find an equation for this area function, they are actually looking for the antiderivative of `f`.

Editor's note, May 2014: For an HTML5 version of the above applet,
click here.

After entering some guesses for an equation of the area function that fits the trace of point `Area`, you can also ask GeoGebra to give you the antiderivative of function `f` by typing the command `Integral[f]`

into the input field. If you drag `f` with the mouse, you will see how its antiderivative is updated automatically too. Again, you are also able to change the function's equation later on, allowing you to examine a variety of different functions by using this single construction.