calculus - Find the slant area of a cone

Question: Find the slant curved area of the surface of revolution of a cone of semi-vertical angle $\alpha$ and base circle of radius a by revolving about the X-axis.

I tried using $ r=a \csc \theta $ and integrating from $\theta=0$ to $ \theta=\alpha$, but the answer is wrong.

Help me with the correct equation and the limits.

Answer

You should be careful to look to what you are integrating.

Consider a differential triangular area in yellow color on the slant side as shown:

$$ dA= \frac12 \frac{a}{\sin \alpha} a \, d \theta $$

Integrating

$$ \int dA = \int_0^{2 \pi} \frac12 \frac{a}{\sin \alpha} a \, d \theta = \frac{\pi a^2}{\sin \alpha}.$$

Also see how the standard slant area formula $ A = \pi a L $ is derived with $ L= \dfrac{a}{\sin \alpha}. $