Antiderivative vs. Integral

Name:

Daniela Yanez 25418161

Tina Sun 58168162

Henry Qiu 50245166

Yifan Jiang 13398169

 

We always think integral and an antiderivative are the same thing. However, I prefer to say that antiderivative is much more general than integral. Specifically, most of us try to use antiderivative to solve integral problems and just view them as the same thing. In fact, further studies imply that there are many situation that we can not apply antiderivative to solve such problems. For example:. When apply antiderivative, we get the answer (by Wolfram Alpha). As the function is not continuous at , it is only an antiderivative but not a integral.

 

An antiderivative refers to a different function whose derivative is the original function. What we mean with that is that if we have a function F'(x)=f(x) in a given interval. For example F(x)=, then F’(x)==f(x). But notice that, where C is a constant, can also be the antiderivative of . So us the illustration shows below jellyfish have same function but slightly different sizes, which represents the constants that each family of antiderivative could have.

 

However, integral can either be the antiderivative of an indefinite integral or an actual number, which means the “anti-derivative” and the “indefinite integral” are the same thing, but the “definite integral” is the area under the curve which is bounded by l and r, which can be actual number.

 

We all learn about Riemann Sum. We can divide the domain into infinite vertical lines (making infinite subintervals) for Riemann integral in one dimension. We can use rectangular area formula to figure out the value.

 

So in some general explanation the main differences are:

Antiderivative talks about undoing differentiation, integrals assign number that tell us how big it is.

A definite integral has limits of integration which give us a number as an answer, while an antiderivative give us a function in terms of the independent variables.

 

It is worth mentioning that there is a Fundamental Theorem of calculus we can use at Riemann integral. It says if G(x) is an antiderivative of g(x). Then we can use G(x) to express the integral of g(x). The Fundamental Theorem of Calculus connect the integral with antiderivative.

But in other dimension( 3-dimension). We can not use this Theorem. We can not find antiderivative of function. At that time, there will be other Theorem we can use———Stokes Theorem and etc.

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