How Do You Calculate the Volume of a Rotated Solid?

[shamieh]

Question: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$x = 2\sqrt{y}, x = 0, y = 9$$

So I know what the graph looks like. But how do i find the points of intersection? wouldn't i just set them equal to each other? so like $$2\sqrt{y} = 9$$ and solve? but how do I solve this equation properly?

 

[alyafey22]

Question: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$x = 2\sqrt{y}, x = 0, y = 9$$

So I know what the graph looks like. But how do i find the points of intersection? wouldn't i just set them equal to each other? so like $$2\sqrt{y} = 9$$ and solve? but how do I solve this equation properly?

You have the equation $$x=2\sqrt{y}$$ and you have the equation $x=0$. Just plug the value of $x=0$ in the equation of the curve. So we have $2\sqrt{y}=0$ and hence by squaring we have $4y=0$ or $y=0$. Hence the curve $x=2\sqrt{y}$ intersects the y-axis at the point $(0,0)$. Similariy find the point of intersection of the two cruves $x=2\sqrt{y}$ and $y=9$.

 

[shamieh]

oh i see! Thanks! so then by setting y = 0 I get 9, then i plug in 9 back to the function and get 2*3 =6, so it intersects at (6,9) ?

 

[alyafey22]

oh i see! Thanks! so then by setting y = 0 I get 9

Why setting the value of $y=0$ ?

, then i plug in 9 back to the function and get 2*3 =6, so it intersects at (6,9) ?

Correct !

 

[Prove It]

Question: Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$$x = 2\sqrt{y}, x = 0, y = 9$$

So I know what the graph looks like. But how do i find the points of intersection? wouldn't i just set them equal to each other? so like $$2\sqrt{y} = 9$$ and solve? but how do I solve this equation properly?

When doing any area or volume of region questions, the first step should ALWAYS be to do a sketch of the region. Then you can at least get an idea of where the intersections are and this can give you an idea on how to refine them.