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Notes

$\text{Properties of Integrals}$
$\underset{―}{\text{Name}}$	$\underset{―}{\text{Formula}}$
$\text{Sum \& Difference}$	${\displaystyle {\int }_a^b(f(x)\pm g(x))dx={\int }_a^{b}f(x)dx\pm {\int }_a^{b}g(x)dx}$
$\text{Constant Multiple}$	${\displaystyle {\int }_a^b(k\cdot f(x))dx=k\cdot {\int }_a^{b}f(x)dx}$
$\text{Order of Interval}$	${\displaystyle {\int }_a^{b}f(x)dx=–{\int }_b^{a}f(x)dx}$
$\text{Zero}$	${\displaystyle {\int }_a^{a}f(x)dx=0}$
$\text{Adding Intervals}$	${\displaystyle {\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx={\int }_a^{c}f(x)dx}$

Questions & Solutions

$\mathbf{\text{1)}}$ ${\displaystyle {\int }_2^{2}f(x)dx}$

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The answer is $0$

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${\displaystyle {\int }_a^{a}f(x)dx=0}$

$\mathbf{\text{2)}}$ ${\displaystyle {\int }_2^{9}3dx}$

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$3(9-2)=21$

$\mathbf{\text{3)}}$ ${\displaystyle {\int }_8^{8}f(x)dx}$

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The answer is $0$

$\mathbf{\text{4)}}$ ${\displaystyle {\int }_1^{10}5dx}$

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$5(10-1)=45$

$\mathbf{\text{5)}}$ ${\displaystyle {\int }_2^{5}f(x)dx=7,{\displaystyle {\int }_5^{9}f(x)dx=4,{\displaystyle {\int }_2^{12}f(x)dx=16}}}$
$\text{Solve for }{\displaystyle {\int }_2^{9}f(x)dx}$

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The answer is $11$

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$\text{Notes}$

${\displaystyle {\int }_a^{c}f(x)dx={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx}$

$\text{Apply notes to this problem.}$

${\displaystyle {\int }_2^{9}f(x)dx=f(x)dx{\displaystyle {\int }_2^{5}f(x)dx+{\displaystyle {\int }_5^{9}f(x)dx}}}$

${\displaystyle {\int }_2^{9}f(x)dx=7+4}$

${\displaystyle {\int }_2^{9}f(x)dx=11}$

$\mathbf{\text{6)}}$ ${\displaystyle {\int }_2^{5}f(x)dx=7,{\displaystyle {\int }_5^{9}f(x)dx=4,{\displaystyle {\int }_2^{12}f(x)dx=16}}}$
$\text{Solve for }{\displaystyle {\int }_5^{12}f(x)dx}$

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${\displaystyle {\int }_2^{12}f(x)dx-{\displaystyle {\int }_2^{5}f(x)dx=16-7=9}}$

$\mathbf{\text{7)}}$ ${\displaystyle {\int }_2^{5}f(x)dx=7,{\displaystyle {\int }_5^{9}f(x)dx=4,{\displaystyle {\int }_2^{12}f(x)dx=16}}}$
$\text{Solve for }{\displaystyle {\int }_9^{5}f(x)dx}$

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$-{\displaystyle {\int }_5^{9}f(x)dx=-4}$

$\mathbf{\text{8)}}$ ${\displaystyle {\int }_2^{5}f(x)dx=7,{\displaystyle {\int }_5^{9}f(x)dx=4,{\displaystyle {\int }_2^{12}f(x)dx=16}}}$
$\text{Solve for }{\displaystyle {\int }_2^{5}4f(x)dx}$

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$4{\displaystyle {\int }_2^{5}f(x)dx=4\cdot 7=28}$

$\mathbf{\text{9)}}$ If${\displaystyle {\int }_a^{b}f(x)dx=5a-2b}$, then what is ${\displaystyle {\int }_a^b[f(x)-3]dx?}$

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$5a-2b-3(b-a)=8a-5b$

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$\text{Notes}$

${\displaystyle {\int }_a^{c}f(x)dx={\int }_a^{b}f(x)dx+{\int }_b^{c}f(x)dx}$

$\text{Apply notes to this problem.}$

${\displaystyle {\int }_a^b[f(x)-3]dx}$

${\displaystyle {\int }_a^{b}f(x)dx–{\displaystyle {\int }_a^{b}3dx}}$

${\displaystyle {\int }_a^{b}f(x)dx–3{\displaystyle {\int }_a^{b}1dx}}$

${\displaystyle 5a-2b–3(b-a)}$

${\displaystyle 5a-2b–3b+3a}$

${\displaystyle 8a-5b}$

See Related Pages$$

$\bullet \text{ Trapezoidal Rule}$
${\displaystyle \frac{b-a}{2n}[f(a)+2f(x_1)+2f(x_2)+\dots +2f{x}_{n-1}+f(b)]\dots }$

$\bullet \text{ Properties of Integrals}$
${\displaystyle {\int }_a^{b}cf(x)dx=c{\displaystyle {\int }_a^{b}f(x)dx\dots }}$

$\bullet \text{ Indefinite Integrals- Power Rule}$
${\displaystyle \int x^{n}dx={\displaystyle \frac{x^{n+1}}{n+1}+C\dots }}$

$\bullet \text{ Indefinite Integrals- Trig Functions}$
${\displaystyle \int \cos xdx=\sin x+C\dots }$

$\bullet \text{ Definite Integrals}$
${\displaystyle {\int }_5^7{x}^{3}dx\dots }$

$\bullet \text{ Integration by Substitution}$
${\displaystyle \int (x^2+3{)}^3(2x)dx\dots }$

$\bullet \text{ Area of Region Between Two Curves}$
$A={\displaystyle {\int }_a^b[f(x)-g(x)]dx\dots }$

$\bullet \text{ Arc Length}$
${\displaystyle {\int }_a^b\sqrt{1+{[f^{\prime }(x)]}^2}dx\dots }$

$\bullet \text{ Average Function Value}$
${\displaystyle \frac{1}{b-a}{\int }_a^{b}f(x)dx}$

$\bullet \text{ Volume by Cross Sections}$
$$

$\bullet \text{ Disk Method}$
$V={\displaystyle {\int }_a^b{[f(x)]}^{2}dx\dots }$

$\bullet \text{ Cylindrical Shells}$
$V=2\pi {\displaystyle {\int }_a^{b}yf(y)dy\dots }$

$\bullet \text{ Andymath Homepage}$

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