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Why is $\int_{-1}^1 x =\frac{x^2}{2}$ rather than $x^2$?

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The integral of $x$ from -1 to 1 is neither of those! Remember that a definite integral (with limits of integration) is a number (think area under the curve) not a function. If you leave off the limits of integration, then you're talking about the antiderivative, and your question makes more sense. – cch Dec 17 '10 at 17:33

First, the integral of $x$ from $-1$ to $1$ is a number, not a function. Definite integrals give you the net signed area between the $X$-axis and the function. So the integral of $x$ from $-1$ to $1$ is neither $x^2$ nor $x^2/2$.

Second, the indefinite integral of $x$ is a family of functions, namely, the family of all functions $F(x)$ such that $F'(x)=x$; that is, the family of antiderivatives of $x$. Since the indefinite integral is a family of functions and not a single function, the indefinite integral of $x$ is neither $x^2/2$ nor $x^2$.

Now, by the Fundamental Theorem of Calculus, if you are trying to compute the definite integral of a function $f(x)$ that is continuous on $[a,b]$, and you are lucky enough to have any antiderivative $F(x)$ of $f(x)$, then you can compute the definite integral as the Total Change of $F(x)$ on $[a,b]$. That is, $$\int_a^b f(x)\,dx = F(b)-F(a).$$

So, in order to compute $$\int_{-1}^1 x\,dx$$ you have several choices:

  1. Find the net signed area geometrically. This is very easy in this case, because all you have is two triangles that cancel each other, so the net signed area is $0$. That is, $\int_{-1}^1 x\,dx = 0$.

  2. Use the definition of the integral as a limit of Riemann sums. Again, not hard to do: the function $y=x$ is increasing, the left hand and right hand sums are easy to compute, and the limit turns out to be $0$. For example, the left hand Riemann sum with $n$ intervals, each of equal length $\frac{2}{n}$, is: \begin{align*} \mathrm{LHS}(n) &= \frac{-n}{n}\left(\frac{2}{n}\right) + \frac{-n+2}{n}\left(\frac{2}{n}\right) + \cdots + \frac{n-2}{n}\left(\frac{2}{n}\right)\\ &= \frac{2}{n^2}\Bigl(-n + (-n+2) + (-n+4)+\cdots+\bigl(-n+(2n-2)\bigr)\Bigr)\\ &= \frac{2}{n^2}\Bigl( -n(n) + 2\bigl(0 + 1 + 2 + \cdots + (n-1)\bigr)\Bigr)\\ &= \frac{2}{n^2}(-n^2 + n^2 - n) = \frac{-2n}{n^2} = -\frac{2}{n}. \end{align*} Thus, $$\lim\limits_{n\to\infty}\mathrm{LHS}(n) = \lim_{n\to\infty}\left(-\frac{2}{n}\right) = 0.$$ The right hand sums also converge to $0$, so the integral is equal to $0$.

  3. Use the Fundamental Theorem of Calculus and find some function $F(x)$ whose derivative is $x$ on $[-1,1]$. Now, $F(x)=x^2$ does not work because $F'(x) = 2x\neq x$. But $\mathcal{F}(x) = \frac{1}{2}x^2$ does work because $\mathcal{F}'(x) = \frac{1}{2}(2x) = x$. That means (by the Constant Function Theorem) that $$\int x\,dx = \frac{1}{2}x^2 + C,$$ (the family of functions that are the antiderivatives of $x$ are the vertical translates of $\frac{1}{2}x^2$) so you can use any one of them (say, the one with $C=0$) to compute $\int_{-1}^1x\,dx$.

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Consider $f(x)=x$ for x equals 0 to a, it is a right-angled triangle (the blue region in figure below). alt text The area of the square (having sides equal a) is $a^2$. Since there are two identical triangles each filling half the area, the area under the curve of f(x)=x for x=0 to a is $\frac{a^2}{2}$, which is precisely what you get from computing the definite integral, since a definite integral calculates area under the curve.

Another example, consider f(x)=x from a to b, with a>=0. The definite integral from a to b calculated the area of the blue region illustrated below. It can be reduced to the previous case as the area of the bigger triangle from 0 to b minus the area of the smaller triangle from 0 to a. Hence $\frac{b^2}{2} -\frac{a^2}{2}$.

alt text

Perhaps I should also mention, when the area is below the x-axis, such as f(x) for x=a to 0 (a<0), a change of sign is needed, i.e $\frac{-a^2}{2}$.

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Integration is the opposite of differentiation.

If you differentiate x^2/2 you get x (you bring down the power of 2, which cancels with the 1/2 out front) which means that when you integrate x you must get x^2/2.

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