Solved: "Evaluate the iterated integral. ∫02∫06x2x3dydx"

remolatg

Answered question

2021-10-29

Evaluate the iterated integral.

\int_{0}^{2} \int_{0}^{6 x^{2}} x^{3} d y d x

yunitsiL

Skilled2021-10-30Added 108 answers

Step 1
To evaluate the integral:

\int_{0}^{2} \int_{0}^{6 x^{2}} x^{3} d y d x

\int_{0}^{2} \int_{0}^{6 x^{2}} x^{3} d y d x = \int_{0}^{2} x^{3} [\int_{0}^{6 x^{2}} d y] d x

= \int_{0}^{2} x^{3} {[y]}_{0}^{6 x^{2}} d x

= \int_{0}^{2} x^{3} [6 x^{2}] d x

= 6 \int_{0}^{2} x^{5} d x

= 6 {[\frac{x^{6}}{6}]}_{0}^{2}

= 6 [\frac{2^{6}}{6} - 0]

= 2^{6}

=64
Step 2
Hence, required answer is 64.

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