Lillie Pittman

2022-07-17

Finding the volume using double integrals

Find the volume of the wedge sliced from the cylinder ${x}^{2}+{y}^{2}=1$ by the planes $z=a(2-x)$ and $z=a(x-2)$.

I am confused because ${x}^{2}+{y}^{2}=1$ is a unit circle not a cylinder. and the other two are lines in the zx plane where $y=0$. I don't see how they are planes, are they planes or are they just lines in the zx plane (when $y=0$).

Find the volume of the wedge sliced from the cylinder ${x}^{2}+{y}^{2}=1$ by the planes $z=a(2-x)$ and $z=a(x-2)$.

I am confused because ${x}^{2}+{y}^{2}=1$ is a unit circle not a cylinder. and the other two are lines in the zx plane where $y=0$. I don't see how they are planes, are they planes or are they just lines in the zx plane (when $y=0$).

lelapem

Beginner2022-07-18Added 12 answers

Explanation:

In a figure showing the (x,z)-plane the two given planes appear as lines intersecting the x-axis at $x=2$ and the z-axis at $z=\pm 2a$. Looking at the figure one realizes that because of symmetry the volume in question is $4a\pi $, whereby $\pi $ stands for the area of the circle ${x}^{2}+{y}^{2}\le 1$.

In a figure showing the (x,z)-plane the two given planes appear as lines intersecting the x-axis at $x=2$ and the z-axis at $z=\pm 2a$. Looking at the figure one realizes that because of symmetry the volume in question is $4a\pi $, whereby $\pi $ stands for the area of the circle ${x}^{2}+{y}^{2}\le 1$.

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