Find out the answer to "Evaluate the following iterated integrals. ∫12∫01(3x2+4y3)dydx"

shadsiei

Answered question

2021-01-10

Evaluate the following iterated integrals.

\int_{1}^{2} \int_{0}^{1} (3 x^{2} + 4 y^{3}) d y d x

irwchh

Skilled2021-01-11Added 102 answers

To evaluate the integral:

\int_{1}^{2} \int_{0}^{1} (3 x^{2} + 4 y^{3}) d y d x

Solution:
When we integrate with respect to one variable then other will be kept as constant.
Evaluating the integral.

\int_{1}^{2} \int_{0}^{1} (3 x^{2} + 4 y^{3}) d y = \int_{1}^{2} [\int_{0}^{1} (3 x^{2} + 4 y^{3}) d y] d x

= \int_{1}^{2} [(3 x^{2} y + 4 \cdot \frac{y^{4}}{4})_{0}^{1}] d x

= \int_{1}^{2} [3 x^{2} \cdot 1 + 1^{4} - 0] d x

= \int_{1}^{2} (3 x^{2} + 1) d x

= [3 \cdot \frac{x^{3}}{3} + x]_{1}^{2}

= [x^{3} + x]_{1}^{2}

= [(2^{3} + 2) - (1^{3} + 1)]

= [12 - 2]

= 10

Hence, required answer is 10

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