Answer: The required area is 131.6626
Step-by-step explanation:
Given that;
equation of the curve; y=x², 3≤ x ≤5
and equation of the surface; f(x,y) = x + √y
Find the Area A = ∫_c ( x + √y) dS
Find the parametrization of the curve.
so let x = a then y = a²
then dx/da = 1, dy/da = 2a
since 3≤ x ≤5 and x = a hence; 3≤ a ≤5
so the area of one side of winding wall will be;
A = ∫_c ( x + √y) dS
= ⁵∫₃ ( a + √a²) √[(dx/da)² + (dy/da)²] da
= ⁵∫₃ ( a + a ) √[(1)² + (2a)²] da
= ⁵∫₃2a√(1 + 4a²) da
= [1/6(4a² + 1 )^3/2)]₃⁵
= - 37/6×√37 + 100/6×√101
= -37.5103 + 169.1729
= 131.6626
Therefore the required area is 131.6626
