Solved: "Evaluate the following iterated integrals. ∫13∫12(y2+y)dx dy"

Cem Hayes

Answered question

2021-01-28

Evaluate the following iterated integrals.

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

faldduE

Skilled2021-01-29Added 109 answers

Let the given integral be I -

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

Integrating with respect to x,

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

I = \int_{1}^{3} (y^{2} + y) (2 - 1) d y

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

Integrating equation with respect to y,

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

I = [\frac{y^{3}}{3} + \frac{y^{2}}{2}]_{1}^{3}

I = (\frac{3^{3}}{3} + \frac{3^{2}}{2}) - (\frac{1^{3}}{3} + \frac{1^{2}}{2})

I = \frac{27}{3} + \frac{9}{2} - \frac{1}{3} - \frac{1}{2}

I = \frac{54 + 27 - 2 - 3}{6}

I = \frac{76}{6}

I = \frac{38}{3}

Hence,

\int_{1}^{3} \int_{1}^{2} (y^{2} + y) d x d y = \frac{38}{3}

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