Final answer:
The surface area generated when the curve y = x is revolved about the y-axis from x = 0 to x = 1 is \(\frac{\pi \sqrt{2}}{2}\) square units.
Step-by-step explanation:
The area of the surface generated by revolving the curve y = x about the y-axis from x = 0 to x = 1 is found using the formula for the surface area of revolution around the y-axis:
\[A = 2\pi \int_{a}^{b} x \sqrt{1 + (\frac{dy}{dx})^2} dx\]
For the curve y = x, the derivative \(\frac{dy}{dx}\) is 1, so the formula simplifies to:
\[A = 2\pi \int_{0}^{1} x \sqrt{2} dx\]
Carrying out the calculation step, we get:
\[A = 2\pi \sqrt{2} \int_{0}^{1} x dx = \pi \sqrt{2} [\frac{1}{2} x^2]_{0}^{1} = \pi \sqrt{2} [\frac{1}{2} (1)^2 - \frac{1}{2} (0)^2]\]
This gives us:
\[A = \pi \sqrt{2} \frac{1}{2} = \frac{\pi \sqrt{2}}{2}\]
The area of the surface generated is \(\frac{\pi \sqrt{2}}{2}\) square units.