SL 5.11—Definite integrals, areas under curve onto x-axis and areas between curves

Imagine you need to find the area of a curved plot of land , not a neat rectangle or triangle, but something with a curved boundary. This is exactly the kind of problem that definite integrals were designed to solve.

Definite Integral: A definite integral is a mathematical tool that calculates the accumulated value of a function over a specific interval $[a,b]$. It is written as:
${\int }_a^{b}f(x)dx$
where $a$ is the lower limit, $b$ is the upper limit, $f(x)$ is the function being integrated, and $dx$ indicates integration with respect to $x$.

Geometrically, the definite integral ${\int }_a^{b}f(x)dx$ represents the net signed area enclosed between the curve $y=f(x)$ and the $x$-axis, from $x=a$ to $x=b$.

• Regions above the $x$-axis contribute positive area.
• Regions below the $x$-axis contribute negative area.

The Fundamental Theorem of Calculus (FTC) is the cornerstone result that links differentiation and integration , two ideas that might seem unrelated at first.

Fundamental Theorem of Calculus: If $F(x)$ is an antiderivative of $f(x)$ (meaning $F^{\prime }(x)=f(x)$), then:
${\int }_a^{b}f(x)dx=F(b)-F(a)$
The definite integral equals the change in the antiderivative evaluated at the upper and lower limits.

This is often written using bracket notation:
${\int }_a^{b}f(x)dx=[F(x){]}_a^b=F(b)-F(a)$

The process for evaluating a definite integral analytically:

1. Find the antiderivative $F(x)$ of $f(x)$ (no constant $+C$ needed for definite integrals).
2. Substitute the upper limit $b$ into $F(x)$.
3. Substitute the lower limit $a$ into $F(x)$.
4. Compute $F(b)-F(a)$.