Calculate the surface area generated by rotating the curve around the x-axis.

Rotate the line

By rotating the line around the x-axis, we generate

a  surface of revolution (a cone without its base.)

Rotate ds

We revolve around the x-axis an element of arc length ds.

This generates a thin strip of area dA.

Surface area of the cone

We get the surface area S of the cone by summing all the elements of area dA as dA sweeps along the complete surface, that is by integrating dA from x = 0 to x = 1.