"Calculate the double integral =∫∫R4x1+xydA,R=[0,4]×[0,1]" was answered by our community

akutagawatj

Answered question

2021-11-26

Calculate the double integral

= \int \int_{R} \frac{4 x}{1 + x y} d A, R = [0, 4] \times [0, 1]

Royce Moore

Beginner2021-11-27Added 17 answers

Step 1
The given integrals is

= \int \int_{R} \frac{4 x}{1 + x y} d A

wrehe

= R = \frac{0, 4}{\times} [0, 1]

Evaluate as follows
Step 2

= \int \int_{R} \frac{4 x}{1 + x y} d A = \int_{0}^{1} \int_{0}^{4} \frac{4 x}{1 + x y} d x d y

= 4 \int_{0}^{1} \int_{0}^{4} (\frac{1}{y} - \frac{1}{y (1 + x y)}) d x d y

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

= 4 \int_{0}^{1} (\frac{4}{y} - \frac{\ln (1 + x y)}{x^{2}}) d y

= 4 \int_{0}^{1} \frac{4}{y^{2}} (4 y - \frac{{\ln (1 + 4 y)}}{4})

= 4 [5 I n (5) - 4]

(by uv meth

= 4 (5 I n (5) - 4)

Thus,

= \int \int_{R} \frac{4 x}{1 + x y} (5 I n (5) - 4)

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