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1. A constant function
The applet shows a graph on the left of the integrand f ' (x) = 2, a constant function. Below is the graph of the antiderivative:
Think of the graph on the right (the antiderivative) as representing the area under the graph on the left (the integrand) from 0 to x. Move the x slider to the right to see the area in green, and the height of the dot on the antiderivative graph, which are the same. In other words, the antiderivative/accumulation function on the right graphs the area under the curve on the left. Note that since the left-hand graph is constant, the area increases linearly with x, so the antiderivative is a line. If you move the x slider so that x < 0, the area becomes negative and is shown in red (for this applet, green area is positive and red area is negative). Move the C slider; what happens to the graph? The accumulation function
will be zero when x = 0, so specifying a specific value for C is like picking what f (0) will be and adding it to the accumulation function.
2. Different slopes
Select the second example from the drop down menu. Now the integrand changes value from -1 to 1 at x = 0. Move the x slider and note the area on the left and the value of the accumulation function/antiderivative on the right. The antiderivative on the right changes from -1 to 1 at x = 0, because the area under the integrand graph switches from positive (above the x axis) to negative (below the x axis). If you move the x slider so that x < 0, the area is still positive (green); why?
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 area on the integrand graph. It starts out increasing at a constant rate, but then the increase in area drops to zero and turns negative. Where does this happen on the right-hand graph?
4. Two slopes
Select the fourth example, showing an integrand made up of two lines. Move the x slider to observe how the area under the integrand (on the left) tells you the value of the antiderivative (on the right).
Other 'Constructing Antiderivatives' topics
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shows the evaluation of a definite integral via the
Fundamental Theorem of Calculus
with constant limits, resulting in a constant value. The "|" notation says to evaluate the expression to the left of the bar at the upper limit, and subtract from that the value of the expression using the lower limit.
has a variable in its upper limit, so using the Fundamental Theorem results in a function as the result. Hence we can think about antiderivatives as being functions that add up area under a curve from some given point.
in which the C's cancel out. This always happens, so it is common not to bother writing them down when evaluating a definite integral using the Fundamental Theorem.
it still means the same thing, because the x in the integrand and the dx, and the x in the upper limit, are different x's. Since this is confusing, it is generally clearer to use different variables for the integrand and the limits when writing out the integral for an accumulation function.