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

\begin{align} \displaystyle \int_{a}^{a}{f(x)}dx &= \big[F(x)\big]_{a}^{a} \\ &= F(a)-F(a) \\ &= 0 \end{align}

### Example 2

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

\begin{align} \displaystyle \int_{b}^{a}{f(x)}dx &= \big[F(x)\big]_{a}^{b} \\ &= F(a)-F(b) \\ &= -\big[F(b)-F(a)\big] \\ &= -\int_{a}^{b}{f(x)}dx \end{align}

### Example 3

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

\begin{align} \displaystyle \int_{b}^{a}{f(x)}dx + \int_{c}^{b}{f(x)}dx &= \big[F(x)\big]_{b}^{a} + \big[F(x)\big]_{c}^{b} \\ &= \big[F(a)-F(b)\big] + \big[F(b)-F(c)\big] \\ &= F(a)-F(c) \\ &= \int_{c}^{a}{f(x)}dx \end{align}

### Example 4

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

\begin{align} \displaystyle \int_{b}^{a}{\big[f(x) + g(x)\big]}dx &= \big[F(x) + G(x)\big]_{b}^{a} \\ &= \big[F(a) + G(a)\big]-\big[F(b) + G(b)\big] \\ &= \big[F(a)-F(b)\big] + \big[G(a)-G(b)\big] \\ &= \int_{b}^{a}{f(x)}dx + \int_{b}^{a}{g(x)}dx \end{align}

### Example 5

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

\begin{align} \displaystyle \int_{b}^{a}{cf(x)}d &= \big[cF(x)\big]_{b}^{a} \\ &= cF(a)-cF(b) \\ &= c\big[F(a)-F(b)\big] \\ &= c \int_{b}^{a}{f(x)}dx \end{align}

### Example 6

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

\begin{align} \displaystyle \int_{b}^{a}{c}dx &= \big[cx\big]_{b}^{a} \\ &= c \times a-c \times b \\ &= c(a-b) \end{align}

### Example 7

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

\begin{align} \displaystyle \int_{1}^{2}{8x^3}dx &= \bigg[\dfrac{8x^{3+1}}{3+1}\bigg]_{1}^{2} \\ &= \bigg[\dfrac{8x^{4}}{4}\bigg]_{1}^{2} \\ &= 2\big[x^4\big]_{1}^{2} \\ &= 2\big[2^4-1^4\big] \\ &= 30 \text{ units}^2 \end{align}

### Example 8

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

\begin{align} \displaystyle \displaystyle \int_{1}^{9}{f(x)}dx &= \displaystyle \int_{1}^{4}{f(x)}dx + \displaystyle \int_{4}^{9}{f(x)}dx \\ &= 10 + 15 \\ &= 25 \end{align}

### Example 9

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

\begin{align} \displaystyle \int_{1}^{-1}{f(x)}dx &= -\int_{-1}^{1}{f(x)}dx \\ &= -5 \end{align}

### Example 10

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

\begin{align} \displaystyle \int_{1}^{-1}{(f(x)+5)}dx &= \int_{1}^{-1}{f(x)}dx +\int_{1}^{-1}{5}dx \\ &= -2 + 5(-1-1) \\ &= -2 + 5 \times (-2) \\ &= -12 \end{align}

### Example 11

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

\begin{align} \displaystyle \int_{-1}^{1}{2f(x)}dx &= 2\int_{1}^{-1}{f(x)}dx \\ &= 2 \times 5 \\ &= 10 \end{align}