The area under a curve

To understand what the definite integral is for, let us pose a concrete problem: computing the area of a region bounded by a curve.

Example — The area under $y = x^4$ between $x=0$ and $x=a$

We want to compute the area of the portion of the plane between the curve $y = x^4$, the $x$-axis and the lines $x=0$ and $x=a$: $A = \int_0^a x^4\,dx = \sum \text{(little rectangles)} = \,?$ How do we compute this sum? Summing “by hand” infinitely many little rectangles is impossible. This is where the antiderivative comes into play, turning this infinite sum into a simple subtraction.

The answer to this problem is developed in the atom Computing the area with the antiderivative.

Try it yourself: increase the number $n$ of rectangles and watch the sum of their areas get closer and closer to the exact area under the curve.

Riemann sums for $f(x) = \tfrac{1}{2}x^2 + 1$ on $[0,4]$: increase $n$ and compare the sum $S(n)$ with the exact value $A = \int_0^4 f(x)\,dx \approx 14.667$.

Links

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