Revise Integration for Cambridge 9709 (1.8)

If we differentiate a function $F(x)$ to get $f(x)$, then the integral of $f(x)$ is $F(x)$. Since the derivative of a constant is zero, we must add a 'constant of integration', denoted by $C$, to our result. This is because $x^2$, $x^2+5$, and $x^2-100$ all have the same derivative, $2x$. Integrating $2x$ gives the family of curves $x^2+C$.

The Power Rule for Integration: $\int ax^n \, dx = \frac{ax^{n+1}}{n+1} + C \quad (\text{for } n \neq -1)$

A definite integral has upper and lower bounds, or 'limits', and evaluates to a single number. This number represents the net area between the function's graph, the x-axis, and the vertical lines defined by the limits. This is one of the most powerful applications of calculus.

The Fundamental Theorem of Calculus: If $\int f(x) \, dx = F(x)$, then the definite integral from $a$ to $b$ is: $\int_a^b f(x) \, dx = [F(x)]_a^b = F(b) - F(a)$

Not all areas are simple regions above the x-axis. Sometimes you need to find the area between two curves, or an area bounded by a curve and the y-axis. For an area between two curves, $f(x)$ and $g(x)$, you integrate the difference: $\int_a^b (\text{upper curve} - \text{lower curve}) \, dx$. For areas bounded by the y-axis, you must rearrange the function to $x=g(y)$ and integrate with respect to $y$ using y-limits.