Finding Volume of Solid Rotated X-Axis

In summary, to find the volume of a solid rotated around the x-axis, you will need to use the formula V = π∫<sup>b</sup><sub>a</sub>f(x)<sup>2</sup> dx, where a and b are the limits of integration and f(x) is the function representing the shape of the solid. This formula is also known as the disk method. The difference between the disk method and the shell method is that the disk method involves slicing the solid perpendicular to the x-axis and summing the volumes of the discs, while the shell method involves slicing the solid parallel to the x-axis and summing the volumes of the shells. The volume of a solid rotated around the x-axis cannot be
  • #1
Cheapo2004
9
0
Ok, I'm supposed to found the volume of the solid that is created after rotating the line f(x) = 2x-1 around the x axis. The limits are y=0 x=3 and x=0. I've been trying for about and hour, and keep getting the answer: 46.0766. I've done the integration tons of times, splitting the problem into two parts for each separate cone, and other stuff. I just can't seem to get the right answer, please help.
 
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  • #2
Tell us how you set up the integral.
 
  • #3


Hi there,

Finding the volume of a solid rotated around the x-axis can be challenging, but with the right approach, it is definitely possible. Let's go through the steps together.

First, we need to visualize the solid that is being rotated. In this case, we have a line f(x) = 2x-1, which looks like a straight line with a y-intercept of -1 and a slope of 2. When we rotate this line around the x-axis, we will get a shape that looks like a cone.

Next, we need to determine the limits of integration. Since we are rotating around the x-axis, the limits will be from x=0 to x=3. This means that we are only interested in the portion of the solid that lies between these two values of x.

Now, we need to set up the integral that will give us the volume of the solid. We can use the formula V = π∫(f(x))^2 dx, where f(x) is the function that defines the shape of the solid and dx is the infinitesimal thickness of the slices that we will be adding up.

Since we are rotating a line around the x-axis, the function f(x) will be the distance from the x-axis to the line at any given value of x. In this case, the distance is simply the value of f(x), which is 2x-1. So, our integral becomes V = π∫(2x-1)^2 dx.

Now, we need to integrate this function from x=0 to x=3. This will give us the volume of the solid between these two values of x. When we do the integration, we get V = π(64/3 - 1/3) = 63π/3 = 66.15.

So, the volume of the solid rotated around the x-axis is approximately 66.15 cubic units. This is different from the answer you got, which was 46.0766. Perhaps there was a mistake in your integration or in setting up the integral. I would recommend double-checking your work to see if you can find where the error occurred.

I hope this helps. Keep practicing and don't get discouraged. Integration can be tricky, but with enough practice, you will get the hang of it. Good luck!
 

1. How do you find the volume of a solid rotated around the x-axis?

To find the volume of a solid rotated around the x-axis, you will need to use the formula V = π∫baf(x)2 dx, where a and b are the limits of integration and f(x) is the function representing the shape of the solid. This formula is also known as the disk method.

2. What is the difference between the disk method and the shell method when finding the volume of a solid rotated around the x-axis?

The disk method involves slicing the solid perpendicular to the x-axis and summing the volumes of the discs to find the total volume. The shell method, on the other hand, involves slicing the solid parallel to the x-axis and summing the volumes of the shells to find the total volume. Both methods can be used to find the volume of solid rotated around the x-axis, but the choice of method depends on the shape of the solid and the ease of integration.

3. Can the volume of a solid rotated around the x-axis be negative?

No, the volume of a solid cannot be negative because it represents the amount of space occupied by the solid. If the resulting volume from the calculation is negative, it is an indication of an error in the calculation or the limits of integration.

4. How do you determine the limits of integration when using the disk method?

The limits of integration for the disk method are determined by the intersection points of the function representing the shape of the solid and the x-axis. These points will be the lower and upper limits of the integral.

5. What is the unit of measurement for the volume of a solid rotated around the x-axis?

The unit of measurement for the volume of a solid rotated around the x-axis will be in cubic units, such as cubic meters or cubic centimeters, depending on the units used for the function representing the shape of the solid and the limits of integration.

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