PLEASE SOLVE IN DIGITSL FORMAT2. Calculate the following double integrals and draw the region ofintegrationr7-3yc) S² S1-³9 12y²3² dxdy

Sol:- ∫12∫17-3y212y-3y2dxdyFirst evaluate ∫17-3y212y-3y2dxIntegral of a constant:…

Answered: PLEASE SOLVE IN DIGITSL FORMAT 2.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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PLEASE SOLVE IN DIGITSL FORMAT
2. Calculate the following double integrals and draw the region of
integration
r7-3y
c) S² S1-³9 12y²3² dxdy

Expert Solution

Step 1

Sol:-

$\int_{1}^{2} \int_{1}^{7 - 3 y} \frac{2}{12 y - 3 y^{2}} d x d y F i r s t e v a l u a t e \int_{1}^{7 - 3 y} \frac{2}{12 y - 3 y^{2}} d x Integral of a constant : \int a d x = a x = {[\frac{2}{12 y - 3 y^{2}} x]}_{1}^{7 - 3 y} Compute the boundaries : \frac{4}{y (- y + 4)} - \frac{2}{- y + 4} = \frac{4}{y (- y + 4)} - \frac{2}{- y + 4} = \frac{2 (- y + 2)}{y (- y + 4)} N o w, = \int_{1}^{2} \frac{2 (- y + 2)}{y (- y + 4)} d y = 2 \cdot \int_{1}^{2} \frac{- y + 2}{y (- y + 4)} d y Expand \frac{- y + 2}{y (- y + 4)} : - \frac{y}{y (- y + 4)} + \frac{2}{y (- y + 4)} Apply the Sum Rule : \int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x = 2 (- \int_{1}^{2} \frac{y}{y (- y + 4)} d y + \int_{1}^{2} \frac{2}{y (- y + 4)} d y) \int_{1}^{2} \frac{y}{y (- y + 4)} d y = \ln (3) - \ln (2) \int_{1}^{2} \frac{2}{y (- y + 4)} d y = \frac{1}{2} \ln (3) = 2 (- (\ln (3) - \ln (2)) + \frac{1}{2} \ln (3)) = - \ln (3) + 2 \ln (2) \int_{1}^{2} \int_{1}^{7 - 3 y} \frac{2}{12 y - 3 y^{2}} d x d y = - \ln (3) + 2 \ln (2) (Decimal : 0.28768 \dots)$