x = 4(3 + y)2
⟹ dx /dy = 4 . 2(3 + y) = 24 + 8y
1+ (dx /dy)2 = 1 + (24 + 8y)2
arc length = ∫14 1+(dx/dy)2 dy = ∫14 1+(24+8y)2 dy
substituting u = 24 + 8y
⟹ du = 8dy , when y =1, u =32 and when y = 4 , u = 56
⟹ arc length = ∫456 1+(u)2 du /8
= [(u1+(u)2 + ln(u +1+(u)2)/16] 456
=[(561+(56)2 + ln(56 +1+(56)2)/16]-[(321+(32)2 + ln(32 +1+(32)2)/16]
=[(563137 + ln(56 +3137)/16]-[(321025 + ln(32 +1025)/16]
=132.0349
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