How Do You Calculate the Volume of a Rotated Solid?

In summary, to find the volume of the solid obtained by rotating the region bounded by the curves x = 2\sqrt{y}, x = 0, y = 9 about a specified line, the first step is to sketch the region. Then, to find the points of intersection, set the equations equal to each other and solve. After finding the points of intersection, use the disk or washer method to find the volume of the solid.
  • #1
shamieh
539
0
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.

\(\displaystyle x = 2\sqrt{y}, x = 0, y = 9\)

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 \(\displaystyle 2\sqrt{y} = 9\) and solve? but how do I solve this equation properly?
 
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  • #2
shamieh said:
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.

\(\displaystyle x = 2\sqrt{y}, x = 0, y = 9\)

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 \(\displaystyle 2\sqrt{y} = 9\) and solve? but how do I solve this equation properly?

You have the equation \(\displaystyle x=2\sqrt{y}\) 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 $y=9$.
 
  • #3
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) ?
 
  • #4
shamieh said:
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 !
 
  • #5
shamieh said:
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.

\(\displaystyle x = 2\sqrt{y}, x = 0, y = 9\)

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 \(\displaystyle 2\sqrt{y} = 9\) 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.
 

FAQ: How Do You Calculate the Volume of a Rotated Solid?

How do you find the intersection point of two lines?

To find the intersection point of two lines, you need to first determine the equations of each line. Then, you can solve the system of equations to find the x and y coordinates of the intersection point. This can be done using various methods such as substitution, elimination, or graphing.

Can a line intersect with more than one other line?

Yes, a line can intersect with multiple other lines. In fact, if you have three or more lines that are not parallel, they will all intersect at a single point.

What if the lines are parallel or coincide?

If the lines are parallel, they will never intersect and therefore there is no intersection point. If the lines coincide (meaning they are the exact same line), then there are infinitely many intersection points, as every point on the line would be an intersection point.

Is there a specific formula for finding the intersection point?

There is not a specific formula for finding the intersection point of two lines, as it depends on the equations of the lines and the method used to solve the system of equations. However, there are general guidelines and steps that can be followed to find the intersection point.

Can this concept be applied to higher dimensions?

Yes, the concept of finding the intersection point of lines can be extended to higher dimensions. In three-dimensional space, for example, you can find the intersection point of two planes by solving a system of three equations. Similarly, in n-dimensional space, you can find the intersection point of n-1 hyperplanes by solving a system of n equations.

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