Find the surface area.

Question

Consider the curve given parametrically by

x(t)=t, y(t)=t² and z(t)=cos .5πt.

What is the area of the surface between the curve and the xy-plane>

Note: below is the link to a graph of the surface.

https://www.desmos.com/3d/edb6d2581f

Answer 1

I was hoping that someone else would answer this question since I am unsure about it.

The surface is symmetric about the z-axis so that the area is given by the line integral over the curve y=x².

The integrand is cos (πt/2) sqrt(1+4t²) dt.

The limits of integration are 0 to 1.

And the integral value must be multiplied by 2.

Wolfram Alpha states that the antiderivative cannot be expressed in terms of elementary functions,

but the value is 2(.824037)=1.64807.