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ediculeN

Answered question

2021-10-25

Evaluate the following integral.

\int_{0}^{2} \int_{x^{2}}^{2 x} x y d y d x

cheekabooy

Skilled2021-10-26Added 83 answers

Step 1:To determine
Evaluate:

\int_{0}^{2} \int_{x^{2}}^{2 x} x y d y d x

Step 2:Calculation
Consider

\int_{0}^{2} \int_{x^{2}}^{2 x} x y d y d x

= \int_{0}^{2} \frac{x y^{2}}{2} ∣_{x^{2}}^{2 x} d x

= \int_{0}^{2} \frac{x}{2} ({(2 x)}^{2} - {(x^{2})}^{2}) d x

= \int_{0}^{2} \frac{x}{2} (4 x^{2} - x^{4}) d x

= \frac{1}{2} \int_{0}^{2} (4 x^{3} - x^{5}) d x

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

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

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

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

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

= \frac{1}{2} (\frac{32}{6})

= \frac{16}{6}

= \frac{8}{3}

Hence,

\int_{0}^{2} \int_{x^{2}}^{2 x} x y d y d x = \frac{8}{3}

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