Integral: area under the graph

Imagine a car pulling away and getting faster and faster. How far has it travelled after ten seconds? If you only know its speed at each moment, the answer hides in the area under the graph. That is what integration is about: adding up many small contributions into a whole.

What does the area under the graph mean?

Draw a function $f(x)$ on a coordinate plane and look at the strip lying between the graph and the x-axis. That strip has an area, and this area is the key quantity. For a horizontal line $f(x) = 3$ the area is simply a rectangle: width times height. For a slanted or curved line the boundary is bent, and we need a trick to find the area.

Summing up thin strips

The trick is old and simple: cut the area into many thin, vertical strips. Each strip is almost a rectangle with width $\Delta x$ and height $f(x)$. Its area is roughly $f(x) \cdot \Delta x$. Adding up all these rectangles gives a good approximation of the whole area. The thinner the strips, the more accurate the approximation.

From the sum to the integral

Now we let the strips become infinitely thin. The sum sign $\sum$ turns into the integral sign $\int$, and $\Delta x$ turns into $dx$. The two numbers $a$ and $b$ at the integral sign say where the area starts and where it ends, they are called the limits of integration. You read the expression as the integral of $f(x)$ from $a$ to $b$.

What the integral measures

When the graph lies above the x-axis, the area counts as positive. When it lies below, it counts as negative. So the integral measures a signed area. For a constant speed the integral gives the distance travelled, because distance is speed times time, and that is exactly the rectangle area under the speed graph. If the car keeps speeding up like the one at the start, the area is no longer a rectangle, but the meaning stays the same: the area under the speed graph is the distance covered.