Integrals

A definite integral computes the net area between a curve $y=f(x)$ and the $x$‑axis from $x=a$ to $x=b$. The fundamental idea is that the integral equals the difference of an antiderivative $F$ evaluated at the endpoints: ${\int }_a^{b}f(x)dx=F(b)-F(a)$. An antiderivative $F$ satisfies $F^{\prime }(x)=f(x)$. For simple power functions $x^n$, the antiderivative is $\frac{x^{n+1}}{n+1}$ (when $n\ne -1$). For $1/x$, the antiderivative is $\ln |x|$. Trigonometric integrands like $\sin x$ integrate to $-\cos x$, and $1/(1+x^2)$ integrates to $\arctan x$. By finding $F$ and plugging in the limits, you obtain the exact area or net signed area without needing to sum rectangles.