Definite Integrals

The Fundamental Theorem of Calculus

For a continuous function $f(x)$ with antiderivative $F(x)$, $$\displaystyle \int_{a}^{b}{f(x)}dx = F(b) - F(a)$$

Properties of Definite Integrals

The following properties of definite integrals can all be deductefd from the fundamental theorem of calculus:
  • $\displaystyle \int_{a}^{a}{f(x)}dx = 0$
  • $\displaystyle \int_{b}^{a}{f(x)}dx = -\int_{a}^{b}{f(x)}dx$
  • $\displaystyle \int_{b}^{a}{f(x)}dx + \int_{c}^{b}{f(x)}dx = \int_{c}^{a}{f(x)}dx$
  • $\displaystyle \int_{b}^{a}{\big[f(x) \pm g(x)\big]}dx = \int_{b}^{a}{f(x)}dx \pm \int_{b}^{a}{g(x)}dx$
  • $\displaystyle \int_{b}^{a}{c}dx = c(a-b)$
  • $\displaystyle \int_{b}^{a}{cf(x)}dx = c\int_{b}^{a}{f(x)}dx$

Example 1

Prove $\displaystyle \int_{a}^{a}{f(x)}dx = 0$.

Example 2

Prove $\displaystyle \int_{b}^{a}{f(x)}dx = -\int_{a}^{b}{f(x)}dx$.

Example 3

Prove $\displaystyle \int_{b}^{a}{f(x)}dx + \int_{c}^{b}{f(x)}dx = \int_{c}^{a}{f(x)}dx$.

Example 4

Prove $\displaystyle \int_{b}^{a}{\big[f(x) + g(x)\big]}dx = \int_{b}^{a}{f(x)}dx + \int_{b}^{a}{g(x)}dx$.

Example 5

Prove $\displaystyle \displaystyle \int_{b}^{a}{cf(x)}dx = c\int_{b}^{a}{f(x)}dx$.

Example 6

Prove $\displaystyle \displaystyle \int_{b}^{a}{c}dx = c(a-b)$.

Example 7

Find $\displaystyle \int_{1}^{2}{8x^3}dx$.

Example 8

If $\displaystyle \int_{1}^{4}{f(x)}dx=10$ and $\displaystyle \int_{4}^{9}{f(x)}dx=15$, find $\displaystyle \int_{1}^{9}{f(x)}dx$.

Example 9

If $\displaystyle \int_{-1}^{1}{f(x)}dx=5$, find $\displaystyle \int_{1}^{-1}{f(x)}dx$.

Example 10

If $\displaystyle \int_{-1}^{1}{f(x)}dx=2$, find $\displaystyle \int_{1}^{-1}{(f(x)+5)}dx$.

Example 11

If $\displaystyle \int_{-1}^{1}{f(x)}dx=5$, find $\displaystyle \int_{-1}^{1}{2f(x)}dx$.