# Calculus/Definite integral/Solutions

$f(x)=x^{6}$
$x=0$
$x=1$
$\left|{\begin{array}{ccccccc}i&x_{i}&x_{i}^{6}&0.2\times x_{i-1}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i-1}^{6}&0.2\times x_{i}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i}^{6}\\0&0.0&0&\,&\,&0&\,\\1&0.2&0.000064&0&0&0.0000128&0.0000128\\2&0.4&0.004096&0.0000128&0.0000128&0.0008192&0.000832\\3&0.6&0.046656&0.0008192&0.000832&0.0093312&0.0101632\\4&0.8&0.262144&0.0093312&0.0101632&0.0524288&0.062592\\5&1.0&1&0.0524288&0.062592&.2&0.262592\end{array}}\right|$
$\left|{\begin{array}{ccccccc}i&x_{i}&x_{i}^{6}&0.2\times x_{i-1}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i-1}^{6}&0.2\times x_{i}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i}^{6}\\0&0.0&0&\,&\,&0&\,\\1&0.2&0.000064&0&0&0.0000128&0.0000128\\2&0.4&0.004096&0.0000128&0.0000128&0.0008192&0.000832\\3&0.6&0.046656&0.0008192&0.000832&0.0093312&0.0101632\\4&0.8&0.262144&0.0093312&0.0101632&0.0524288&0.062592\\5&1.0&1&0.0524288&0.062592&.2&0.262592\end{array}}\right|$
$f(x)=x^{6}$
$x=1$
$x=2$
$\left|{\begin{array}{ccccccc}i&x_{i}&x_{i}^{6}&0.2\times x_{i-1}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i-1}^{6}&0.2\times x_{i}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i}^{6}\\0&1.0&1&\,&\,&.2&\,\\1&1.2&2.985984&.2&.2&0.5971968&0.5971968\\2&1.4&7.529536&0.5971968&0.7971968&1.5059072&2.103104\\3&1.6&16.777216&1.5059072&2.303104&3.3554432&5.4585472\\4&1.8&34.012224&3.3554432&5.6585472&6.8024448&12.260992\\5&2.0&64&6.8024448&12.460992&12.8&25.060992\end{array}}\right|$
$\left|{\begin{array}{ccccccc}i&x_{i}&x_{i}^{6}&0.2\times x_{i-1}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i-1}^{6}&0.2\times x_{i}^{6}&\sum \limits _{k=1}^{i}0.2\times x_{i}^{6}\\0&1.0&1&\,&\,&.2&\,\\1&1.2&2.985984&.2&.2&0.5971968&0.5971968\\2&1.4&7.529536&0.5971968&0.7971968&1.5059072&2.103104\\3&1.6&16.777216&1.5059072&2.303104&3.3554432&5.4585472\\4&1.8&34.012224&3.3554432&5.6585472&6.8024448&12.260992\\5&2.0&64&6.8024448&12.460992&12.8&25.060992\end{array}}\right|$
$f(x)=x$
$g(x)=x^{2}$
$x=0$
$x=1$
$\int _{0}^{1}xdx={\frac {1}{2}}$
$\int _{a}^{b}x^{2}dx={\frac {b^{3}}{3}}-{\frac {a^{3}}{3}}$
$\int _{0}^{1}(x-x^{2})dx={\frac {1}{2}}-({\frac {1^{3}}{3}}-{\frac {0^{3}}{3}})={\frac {1}{2}}-{\frac {1}{3}}=\mathbf {\frac {1}{6}}$
$\int _{0}^{1}xdx={\frac {1}{2}}$
$\int _{a}^{b}x^{2}dx={\frac {b^{3}}{3}}-{\frac {a^{3}}{3}}$
$\int _{0}^{1}(x-x^{2})dx={\frac {1}{2}}-({\frac {1^{3}}{3}}-{\frac {0^{3}}{3}})={\frac {1}{2}}-{\frac {1}{3}}=\mathbf {\frac {1}{6}}$
$\int _{0}^{2}x^{6}dx$
    $0.062592<\int _{0}^{1}x^{6}dx<0.262592$
and in exercise 2 we found that
    $12.460992<\int _{1}^{2}x^{6}dx<25.060992$
From this we can deduce that
    $0.062592+12.460992<\int _{0}^{1}x^{6}dx+\int _{1}^{2}x^{6}dx<0.262592+25.060992$
    $\mathbf {12.523584<\int _{0}^{2}x^{6}dx<25.323584}$
    $0.062592<\int _{0}^{1}x^{6}dx<0.262592$
and in exercise 2 we found that
    $12.460992<\int _{1}^{2}x^{6}dx<25.060992$
From this we can deduce that
    $0.062592+12.460992<\int _{0}^{1}x^{6}dx+\int _{1}^{2}x^{6}dx<0.262592+25.060992$
    $\mathbf {12.523584<\int _{0}^{2}x^{6}dx<25.323584}$
$f$
$a$
    $\int _{-a}^{a}f(x)dx=2\int _{0}^{a}f(x)dx.$
    $\int _{-a}^{a}f(x)dx=\int _{-a}^{0}f(x)dx+\int _{0}^{a}f(x)dx$
Make the substitution $u=-x;du=-dx$. $u=a$ when $x=-a$ and $u=0$ when $x=0$. Then
    $\int _{-a}^{0}f(x)dx=\int _{a}^{0}f(-u)(-du)=-\int _{a}^{0}f(-u)du=\int _{0}^{a}f(-u)du=\int _{0}^{a}f(u)du$
where the last step has used the evenness of $f$. Since $u$ is just a dummy variable, we can replace it with $x$. Then
    $\int _{-a}^{a}f(x)dx=\int _{0}^{a}f(x)dx+\int _{0}^{a}f(x)dx=2\int _{0}^{a}f(x)dx$
    $\int _{-a}^{a}f(x)dx=\int _{-a}^{0}f(x)dx+\int _{0}^{a}f(x)dx$
Make the substitution $u=-x;du=-dx$. $u=a$ when $x=-a$ and $u=0$ when $x=0$. Then
    $\int _{-a}^{0}f(x)dx=\int _{a}^{0}f(-u)(-du)=-\int _{a}^{0}f(-u)du=\int _{0}^{a}f(-u)du=\int _{0}^{a}f(u)du$
where the last step has used the evenness of $f$. Since $u$ is just a dummy variable, we can replace it with $x$. Then
    $\int _{-a}^{a}f(x)dx=\int _{0}^{a}f(x)dx+\int _{0}^{a}f(x)dx=2\int _{0}^{a}f(x)dx$
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