![]() Now, all of a sudden we have a reason to call the indefinite integral the antiderivative. This means that the integral really is undone by the derivative…ie. If you take the derivative of a definite integral from to with respect to, it gives back the function that you are integrating. The key take-home point however is the following: It starts off small, then it peaks around 1, then it gets less steep again by 1.5. Think about the gradient of the red curve. the red curve is an antiderivative of the blue curve. The fundamental theorem of calculus then says: The red curve is the function whose gradient is the blue curve. Of course because the area is always above the axis, this is just continuously increasing (as we increase we are just adding more area to it). At the same time we are plotting, in red, the total area from 0 up to. We are varying the upper limit up to which we are looking for the area (you see the right hand side of the region moving further to the right). We are looking at the area under this curve which is the region shaded in blue. We have a graph in blue of some function. This needs to be digested properly, but first I’ll give you a little animation
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