ediculeN

2021-10-25

Evaluate the following integral.

${\int}_{0}^{2}{\int}_{{x}^{2}}^{2x}xydydx$

cheekabooy

Skilled2021-10-26Added 83 answers

Step 1:To determine

Evaluate:

${\int}_{0}^{2}{\int}_{{x}^{2}}^{2x}xydydx$

Step 2:Calculation

Consider${\int}_{0}^{2}{\int}_{{x}^{2}}^{2x}xydydx$

$={\int}_{0}^{2}\frac{x{y}^{2}}{2}{\mid}_{{x}^{2}}^{2x}dx$

$={\int}_{0}^{2}\frac{x}{2}({\left(2x\right)}^{2}-{\left({x}^{2}\right)}^{2})dx$

$={\int}_{0}^{2}\frac{x}{2}(4{x}^{2}-{x}^{4})dx$

$=\frac{1}{2}{\int}_{0}^{2}(4{x}^{3}-{x}^{5})dx$

$=\frac{1}{2}(\frac{4{x}^{4}}{4}-\frac{{x}^{6}}{6}){\mid}_{0}^{2}$

$=\frac{1}{2}({x}^{4}-\frac{{x}^{6}}{6}){\mid}_{0}^{2}$

$=\frac{1}{2}({2}^{4}-\frac{{2}^{6}}{6}-0+0)$

$=\frac{1}{2}(16-\frac{64}{6})$

$=\frac{1}{2}\left(\frac{96-64}{6}\right)$

$=\frac{1}{2}\left(\frac{32}{6}\right)$

$=\frac{16}{6}$

$=\frac{8}{3}$

Hence,$\int}_{0}^{2}{\int}_{{x}^{2}}^{2x}xydydx=\frac{8}{3$

Evaluate:

Step 2:Calculation

Consider

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