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