Loriezon Claveria

2022-07-13

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Jeffrey Jordon

Expert

Remove parentheses.

${\int }_{0}^{9}{\int }_{\frac{y}{3}}^{\sqrt{3}}x{y}^{2}dxdy$

Evaluate ${\int }_{\frac{y}{3}}^{\sqrt{3}}x{y}^{2}dx$.

Since ${y}^{2}$ is constant with respect to $x$, move ${y}^{2}$ out of the integral.

${\int }_{0}^{9}{y}^{2}{\int }_{\frac{y}{3}}^{\sqrt{3}}xdxdy$

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2}{x}^{2}$.

${\int }_{0}^{9}{y}^{2}\frac{1}{2}{x}^{2}{\right]}_{\frac{y}{3}}^{\sqrt{3}}dy$

${\int }_{0}^{9}{y}^{2}\left(\frac{3}{2}-\frac{1}{2}{\left(\frac{1}{3}y\right)}^{2}\right)dy$

Simplify.

${\int }_{0}^{9}\frac{3{y}^{2}}{2}-\frac{{y}^{4}}{18}dy$

Evaluate ${\int }_{0}^{9}\frac{3{y}^{2}}{2}-\frac{{y}^{4}}{18}dy$.

Split the single integral into multiple integrals.

${\int }_{0}^{9}\frac{3{y}^{2}}{2}dy+{\int }_{0}^{9}-\frac{{y}^{4}}{18}dy$

Since $\frac{3}{2}$ is constant with respect to $y$, move $\frac{3}{2}$ out of the integral.

$\frac{3}{2}{\int }_{0}^{9}{y}^{2}dy+{\int }_{0}^{9}-\frac{{y}^{4}}{18}dy$

By the Power Rule, the integral of ${y}^{2}$ with respect to $y$ is $\frac{1}{3}{y}^{3}$.

3213y3]90+∫90−y418dy$\frac{3}{2}\frac{1}{3}{y}^{3}{\right]}_{0}^{9}+{\int }_{0}^{9}-\frac{{y}^{4}}{18}dy$

Since $-1$ is constant with respect to $y$, move $-1$ out of the integral.

$\frac{3}{2}\frac{1}{3}{y}^{3}{\right]}_{0}^{9}-{\int }_{0}^{9}\frac{{y}^{4}}{18}dy$

Since $\frac{1}{18}$ is constant with respect to $y$, move $\frac{1}{18}$ out of the integral.

$\frac{3}{2}\frac{1}{3}{y}^{3}{\right]}_{0}^{9}-\left(\frac{1}{18}{\int }_{0}^{9}{y}^{4}dy\right)$

By the Power Rule, the integral of ${y}^{4}$ with respect to $y$ is $\frac{1}{5}{y}^{5}$.

$\frac{3}{2}\frac{1}{3}{y}^{3}{\right]}_{0}^{9}-\frac{1}{18}\frac{1}{5}{y}^{5}{\right]}_{0}^{9}$

Substitute and simplify.

Evaluate $\frac{1}{3}{y}^{3}$ at $9$ and at $0$.

$\frac{3}{2}\left(\left(\frac{1}{3}\cdot {9}^{3}\right)-\frac{1}{3}\cdot {0}^{3}\right)-\frac{1}{18}\frac{1}{5}{y}^{5}{\right]}_{0}^{9}$

Evaluate $\frac{1}{5}{y}^{5}$ at $9$ and at $0$.

$\frac{3}{2}\left(\frac{1}{3}\cdot {9}^{3}-\frac{1}{3}\cdot {0}^{3}\right)-\frac{1}{18}\left(\frac{1}{5}\cdot {9}^{5}-\frac{1}{5}\cdot {0}^{5}\right)$

Simplify.

$-\frac{1458}{5}$

The result can be shown in multiple forms.

Exact Form:

$-\frac{1458}{5}$

Decimal Form:

$-291.6$

Mixed Number Form:

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