for example,
doesn’t give a definite value of the answer to the question, why is that?
On the other hand, the same question, except rotating along y axis, gives an answer: volume of revolved solid of region bounded by y = -2x + 1 , y = 5, x = 0 about the y axis - Wolfram|Alpha
This is creating difficulties, and any help is deeply appreciated.
Neither the first nor the second input give a value to me, but
In[41]:= (* x Rotation *)
\[Pi] Integrate[(-2 x + 1)^2, {x, -2, 0}]
Out[41]= (62 \[Pi])/3
In[42]:= (* y Rotation *)
\[Pi] Integrate[(-1/2 (y - 1))^2, {y, 0, 1}] + \[Pi] Integrate[(1/2 (y - 1))^2, {y, 1, 5}]
Out[42]= (65 \[Pi])/12
because in cylindrical co-ordinates the volume is \int_a^b\int_0^{2\pi}\int_0^{f(z)} r dr d\varphi dz = 2\pi \int_a^b \frac{1}{2} (f(z))^2 dz = \pi \int_a^b (f(z))^2dz where f(z) is understood to be non-negative, i.e. if f(z) has a zero one has to stop integration at it and continue with the next positive part of the boundary description becoming f(z) and so on.