Volume computation not returning expected result

Question

I have tried to integrate the expression

(0.5*(3*Cos[t]*Cos[t]-1))*(0.5*(3*Cos[t]*Cos[t]-1))

where t is in [0, π], as a triple integral to get volume. The same function is visualized as shown in the figure below. I know that the surface is enclosed in a box of dimension 0.5 x 0.5 x 2.0, so the box's volume is 0.50 (Unit^3),

However, when I integrate the expression with various methods, I have got,

V = 0.837758 (Unit^3 !)

or

NIntegrate[
  (1/2)*(1/2)*(3*Cos[t]*Cos[t] - 1)*(3*Cos[t]*Cos[t] - 1)*r*r*Sin[t], 
  {r, 0, 1}, {t, 0, Pi}, {p, 0, 2*Pi}]

= 0.837758

My point is: It can be clearly seen that the surface is well encapsulated in the box, so its volume is expected < 0.50 Unit^3, but I'm not getting that result.

So I am confused here. Can anyone help me to figure what went wrong?

Answer 1

(1/2)*(1/2)*(3*Cos[t]*Cos[t] - 1)*(3*Cos[t]*Cos[t] - 1) is not the density, it is the maximal radius of the body. So you evaluate the wrong integral. Try this:

NIntegrate[
 r r Sin[t], 
 {t, 0, Pi}, 
 {p, 0, 2*Pi}, 
 {r, 0, 1/4 (3 Cos[t] Cos[t] - 1)^2}
 ]

0.221784