Integration
Integration
- Integration reverses differentiation:
$$\int (ax+b)^n\, dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C \quad (n \neq -1)$$
- The $+\,C$ is the constant of integration — find it from a known point.
Practice
Why does an indefinite integral include "+ C"?
Differentiating a constant gives 0, so the original could have had any constant — hence + C.
Definite integrals & volume
- A definite integral gives a number: $\displaystyle\int_p^q f(x)\,dx = F(q) - F(p)$.
- The area under a curve from $p$ to $q$ is $\displaystyle\int_p^q y\,dx$ (for two curves, integrate top − bottom).
- Volume of revolution about the $x$-axis: $V = \pi\displaystyle\int_p^q y^2\,dx$.
Practice
Evaluate the definite integral of 2x from 0 to 3 (∫₀³ 2x dx).
∫2x dx = x², so [x²]₀³ = 9 − 0 = 9.
Practice
Evaluate ∫₀² 3x² dx.
∫3x² dx = x³, so [x³]₀² = 8 − 0 = 8.
You've got it
Key idea
- integration reverses differentiation; remember the $+\,C$
- a definite integral $\int_p^q f(x)\,dx = F(q) - F(p)$ = the area under the curve
- volume of revolution about the $x$-axis $= \pi\int y^2\,dx$