A simple integral

In A simple derivative we played with the derivative of functions of the form xⁿ.

Here we play with those same functions, but now we explore the (definite) integral of those functions.

The integral consist of the 'area under the curve' of a function across an interval from x = a to x = b.

The Wikipedia page on integrals has a nice image of this:



We compute this integral by taking the so-called anti-derivative of the function and subtracting its value at a from that of the value at b.

The antiderivative itself is the function the derivative of which is the original function:

A few examples:

• f(x) = x —> ∫x dx = (1/2) x^2*
• f(x) = x² —> ∫x² dx = (1/3) x³
• f(x) = xⁿ —> ∫x dx = 1/(n + 1) x^{(n + 1)}

Below you can play with this. Select the exponent n for the function f(x) = xⁿ and two values of x between which the integral is to be computed.

The browser will compute the integral and plot the function as well as the corresponding area under the curve.

n (-3 ≤ n ≤ 3) in xn: Low x value (-10 ≤ x ≤ 10):    High x value (-10 ≤ x ≤ 10):    Plot!

^*You may want to compare this integral with computing the area of a triangle. After all, the area under the curve of the function f(x) = x where 0 ≤ x ≤ b forms a right triangle the area of which equals b × b / 2.