Explained: "Evaluate the iterated integral. ∫13∫0y4x2+y2dxdy"

sanuluy

Answered question

2021-10-17

Evaluate the iterated integral.

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

Talisha

Skilled2021-10-18Added 93 answers

Step 1
Consider the following integral:

\int_{1}^{3} \int_{0}^{y} \frac{4}{x^{2} + y^{2}} d x d y = I

Consider the following formula:

\int \frac{1}{x^{2} + a^{2}} d x = \frac{1}{a} \arctan (\frac{x}{a}) + C

I = 4 \int_{1}^{3} {[\frac{1}{y} \arctan (\frac{x}{y})]}_{0}^{y} d y

= 4 \int_{1}^{3} [\frac{1}{y} \arctan (\frac{y}{y}) - \frac{1}{y} \arctan (\frac{0}{y})] d y

= 4 \int_{1}^{3} \frac{π}{4 y} d y

= \int_{1}^{3} \frac{π}{y} d y

Step 2
Consider the following formula:

\int \frac{1}{y} d y = \ln | y | + C

I = π {[\ln | y |]}_{1}^{3}

= π [\ln (3) - \ln (1)]

= π \ln (3)

Step 3
Hence, the solution is

π \ln (3)

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