Computing the area with the antiderivative

Let us take up the problem left pending: the area under $y=x^4$ between $x=0$ and $x=a$. Now we have the tool to solve it.

Remark — Completing the example

For $f(x) = x^4$ we look for $F$ with $F'(x) = x^4$. The answer is $F(x) = \dfrac{x^5}{5}$, because $\left(\dfrac{x^5}{5}\right)' = x^4$. Hence: $A = \int_0^a x^4\,dx = F(a) - F(0) = \frac{a^5}{5} - 0 = \boxed{\dfrac{a^5}{5}}.$

The infinite sum of little rectangles has been reduced to a simple calculation: find the antiderivative and evaluate it at the endpoints. This is the general method with which we will compute all definite integrals.

Links

Topics: Integral
Concepts: Area beneath a curve · Definite integral · Antiderivative
Skills: Calculating · Integrating