Question

4. Sketch the domain of integration of I = order of integration. VT dy dr. Evaluate I by changing the (20p)

Answer 1

1 I Tyan +ndyda

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ISA VI 2

C

h 5 xso

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I 1 -dardy y4 +1

=$1 [] 1 -dy y +1

$=\int_{0}^{1}[y^3-0]\frac{1}{y^4+1}dy$

$=\int_{0}^{1}\frac{y^3}{y^4+1}dy$

Putting

$y^4+1=t$

$4y^3dy=dt$

$y^3dy=dt/4$

(1/9), I

$=\frac{1}{4}\int_{1}^{2}\frac{1}{t}dt$

$=\frac{1}{4} [\ln t ]_{1}^{2}$

$=\frac{1}{4} [\ln 2-\ln 1 ]$

$\boldsymbol{=\frac{\ln 2}{4} }$