Concept ExplanationWhat are anti-derivatives? What are integrals? Some examples are not neccesary, but appreciated.  

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embizze eNotes educator| Certified Educator

(1) Basically, an antiderivative of a function is another function whose derivative is the given function.

e.g. An antiderivative of `f(x)=x^2` is `F(x)=1/3x^3` since `d/(dx)F(x)=f(x)` . However, this is not completely correct since there are other functions whose derivatives are `f(x)=x^2` such as `1/3x^3-1,1/3x^3+pi` , etc... Any function of the form `F(x)=1/3x^3+C` where `C` is a constant will have derivative `f(x)=x^2` .

This is true of all functions with antiderivatives -- if you find one antiderivative `F(x)` , then all other antiderivatives are of the form `F(x)+C` .

The notation for finding an antiderivative is `int f(x)dx=F(x)+C` , an indefinite integral.

(2) A definite integral, on the other hand, is a number. The number is usually defined as the Riemann sum for the given function on the interval. The notation is `int_a^bf(x)dx` .

e.g. `int_0^1x^2dx=1/3x^3|_0^1=1/3-0=1/3` . Since this function is positive on the interval, this is equivalent to the area between the curve and the x-axis.

Note that in computing the definite integral we made use of the antiderivative. This is the fundamental theorem of calculus.

Thus the key difference between a definite integral and an antiderivative is that the first results in a number (sometimes associated with area with suitable restrictions) and the second in a family of functions.


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