(865, #13) Find the exact area of the surface z = x² + y², 0 <= x <= 1, 0 <= y <= 1.

Solution: The surface is constructed by sliding the parabola z = x² along the line z = 2y:

plot3d( { x^2 + 2*y, 0}, x = 0..1, y = 0..1, grid =[10,6],
color = red, tickmarks = [2,2,2], axes = framed,
labelfont = [HELVETICA, BOLD, 10]);

As for its area:

Integrating with respect to y we get

Use substitution. Set

Then

By Maple, the exact surface area is 3/2+5/8*ln(5):

lprint((5/2)*int(sqrt(u^2+1),u=0..2/sqrt(5)));

3/2+5/8*ln(5)

by: gm