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The '''indefinite integral''' or '''antiderivative''' of a function <math>f(x) \ </math> is any function <math>g(x) \ </math> such that <math>\frac{d}{dx}g(x) = f(x) \ </math>.

The '''indefinite integral''' or '''antiderivative''' of a function <math>f(x) \ </math> is any function <math>g(x) \ </math> such that <math>\frac{d}{dx}g(x) = f(x) \ </math>.

The '''definite integral''', or as it is defined in elementary calculus, the '''Riemann sum''', is a quantity ''EXPAND THIS''

The '''definite integral''' is a very important concept for students wishing to learn multivariate calculus, as we define many different kinds of definite integrals in this course. 

 

Let <math>f(x) \ </math> be any function associating real numbers with real numbers, and let the [[Closed set|closed interval]] <math>[a,b] \ </math> be in the domain of the function.  Let's divide this interval into <math>n \ </math> smaller intervals, so that the first one is <math>[a,q_1] \ </math> and the last one is <math>[q_{n-1},b] \ </math>, and for convenience we'll call <math>a = q_0, b=q_n \ </math>.  Furthermore, let's pick a point in each of these intervals, say <math>x_j \in [q_{n-1},q_n] \ </math>.  Now we form the sum

 

 

This is called a '''Riemann sum''', named after the famous 19th century mathematician [[Bernhard Reimann]].  If this sum approaches a limit as <math>n \rightarrow \ </math>[[Infinity|<math>\infty \ </math>]] no matter how we choose our <math>q_j \ </math>'s or our <math>x_j \ </math>'s, then <math>f \ </math> is said to be '''Riemann integrable''' and that limit is called the definite integral of <math>f \ </math> from <math>a \ </math> to <math>b \ </math>, and this is written

 

The '''fundamental theorem of calculus''' states that <math>\int_{x_o}^x \frac{d}{dx}f(x)dx = f(x) \ </math>.

The '''fundamental theorem of calculus''' states that <math>\int_{x_o}^x \frac{d}{dx}f(x)dx = f(x) \ </math>.