So I've just got my cheese from the cheesemonger and he's cut it along the axis $x$,$y$, and $z$ so I can do my math homework with it. The lengths of my cheese are:
$x$=6
$y$=4
$z$=2
I'm asked to find the volume using calculus, because using geometry would be cheating, evidently.
The three ways I am asked to "cut up" the cheese are perpendicular to the $y$-axis (triangles), the $x$-axis (rectangles), and $z$-axis (also rectangles).
This is easy, and it turns out the volume is 24 units cubed (all three times). E.g:
4$\int_0^6 \frac{x}{3} \mathrm{d}x = 24$
Now, I ask myself what if I want to take the other rectangle, and use that as my slice. I know that width is 4 ($y$-axis), and the length is z$\sqrt{10}$ (pythagoras with $\sqrt{z^2 + (3z)^2}$). But how do I find my limits of integration?
$4\sqrt{10}\int_a^b z \mathrm{d}z.$
I would need to know the distance from the $y$-axis to the top (in this photo) of the wedge. However, I've exhausted my math knowledge here.
I asked my calc teacher, and he told me that I find this number by "taking more calculus classes" (a good joke actually). Since I don't have time to go bug him in his office hours, can someone explain some of the different ways to find this distance?