Calculating the volume and lateral area of cones

[TheBaker]

This is probably just me being a bit of an idiot (I'm going to blame exam stress), but why do the following ways of calculating the volume and lateral area of cones produce different results?

I'll use the following equation of a cone to demonstrate:

$$x^2 + y^2 = \frac{9}{4}z^2$$ (Valid for $$0 \leq z \leq 1$$)

Volume - Summing Disks

$$V = \int_0^1 \pi r^2 dz = \int_0^1 \frac{9\pi}{4}z^2 dz = \frac{3\pi}{4}$$

This is the correct answer. However, if instead of summing disks from bottom to top, I sum triangles around the cone...

Volume - Summing Triangles

$$A = \frac{1}{2} \mbox{base} \times \mbox{height} = \frac{1}{2} \frac{3}{2}$$

Then, integrating this around a circle to form a cone I get:

$$\int_0^{2\pi} \frac{3}{4} d\theta = \frac{3 \pi}{2}$$

This is twice as large as the (correct) volume found the other way, but conceptually I can't see where I've made a mistake. A similar thing happens for the surface area - when you sum the hypotenuse of the triangle or the circumference of the disks.

 

[HallsofIvy]

The distance through which a triangle "sweeps" when moved by an angle $d\theta$ is not constant- in particular, it is not $d\theta$ (which is not a distance to begin with). It varies with distance from the z-axis as $r d\theta$.

 

[TheBaker]

Ah, gotcha. Thanks.