Find the volume of the solid bounded by the cylinder $x^2 + y^2 = 1$ and the planes $x + y + z = 1$ and $4x + 2y + z = 7$.
I am able to find the radius of the cylinder by switching to cylindrical coordinates.
Let $x = \rho \cos(\theta)$, $y = \rho \sin(\theta)$, $z = \zeta$ $\forall \rho \ge 0, \theta \in [0, 2\pi], \zeta \in \mathbb{R}$.
$x^2 + y^2 = 1$ in $\mathbb{R}^3$ is a cylinder of radius $1$.
$\therefore \rho^2 \cos^2(\theta) + \rho^2\sin^2(\theta) = 1$
$\implies \rho^2 = 1$
$\therefore 0 \le \rho \le 1$ since $\rho \ge 0$
But this is only within the context of the cylinder. Usually, I would find the intersection points between the objects and then find the limits of integration. However, in this case, I can't find a way to equate the planes with the cylinder and algebraically manipulate the equation to find intersection points and limits of integration. Of course, we could find all of these graphically by drawing a diagram, but, for the purpose of generality, I am interested in solving these problems analytically.
I'm out of ideas and unsure of how to proceed with calculations in such a way that will enable me to analytically determine the intersection points and limits between these 3 objects. I would greatly appreciate it if people could please take the time to demonstrate how to go about this and explain the reasoning behind each step so that I may learn to do it for myself in the future.
I think I've managed to solve this problem analytically.
We want to find the volume of the solid bounded by the cylinder $x^2 + y^2 = 1$ and the planes $x + y + z = 1$ and $4x + 2y + z = 7$.
Since we are dealing with a cylinder, we will convert to cylindrical coordinates.
Let $x = \rho \cos(\theta)$, $y = \rho \sin(\theta)$, $z = \zeta$ $\forall \rho \ge 0, \theta \in [0, 2\pi], \zeta \in \mathbb{R}$.
$x^2 + y^2 = 1$ in $\mathbb{R}^3$ is a cylinder of radius $1$.
$\therefore \rho^2 \cos^2(\theta) + \rho^2\sin^2(\theta) = 1$
$\implies \rho^2 = 1$
$\therefore 0 \le \rho \le 1$ since $\rho \ge 0$
We now know that $\theta \in [0, 2\pi], \rho \in [0, 1]$. Therefore, the only bounds that remain to be found are those for $\zeta$.
$x + y + z = 1$
$\therefore \rho \cos(\theta) + \rho \sin(\theta) + \zeta = 1$
$\implies \zeta = 1 - \rho \cos(\theta) - \rho \sin(\theta)$
$4x + 2y + z = 7$
$\therefore 4\rho \cos(\theta) + 2\rho \sin(\theta) + \zeta = 7$
$\implies \zeta = 7 - 4\rho \cos(\theta) - 2\rho \sin(\theta)$
Therefore, the bounds for $\zeta$ are $\zeta = 7 - 4\rho \cos(\theta) - 2\rho \sin(\theta)$ and $\zeta = 1 - \rho \cos(\theta) - \rho \sin(\theta)$.
However, we still need to know which value of $\zeta$ is the upper limit and which is the lower limit. If we draw a diagram, then we will be able to see that $\zeta = 7 - 4\rho \cos(\theta) - 2\rho \sin(\theta)$ is the upper limit. But how can we find this analytically?
We know that $\rho \in [0, 1]$, $\theta \in [0, 2\pi]$.
If $\rho = 0$, then $\zeta = 1$ and $\zeta = 7$.
If $\rho = 1$ and $\theta = 0$, then $\zeta = 0$ and $\zeta = 3$.
If $\rho = 1$ and $\theta = \dfrac{\pi}{2}$, then $\zeta = 0$ and $\zeta = 5$.
If $\rho = 1$ and $\theta = \pi$, then $\zeta = 2$ and $\zeta = 11$.
If $\rho = 1$ and $\theta = \dfrac{3\pi}{2}$, then $\zeta = 2$ and $\zeta = 9$.
$\therefore D = \{(\rho, \theta, \zeta) \mid 0 \le \rho \le 1, 0 \le \theta \le 2\pi, 1 - \rho \cos(\theta) - \rho \sin(\theta) \le \zeta \le 7 - 4\rho \cos(\theta) - 2\rho \sin(\theta) \}$
$\int_{\theta = 0}^{2\pi} \int_{\rho = 0}^{1} \int_{1 - \rho \cos(\theta) - \rho \sin(\theta)}^{7 - 4\rho \cos(\theta) - 2\rho \sin(\theta)} (1 \cdot \rho) d\zeta d\rho d\theta$ where $\rho$ in the integrand is the scaling factor for the linear transformation from xy-coordinates to cylindrical coordinates.
$= 6\pi$