Finding Volume by Rotating Curves around the x-axis

In summary, the task is to find the volume of a solid formed by rotating the region bounded by the curves y = 6x^2, y = 0, and x = 1 around the x-axis. The formula for finding volume is V = A(x)*h, where h = dx. The attempt at a solution involved using 6x^2 as the radius, squaring it, multiplying it by π, and integrating it using dx as the height. However, the mistake was made in approaching the problem and the correct solution was not found.
  • #1
EngnrMatt
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Homework Statement



Find the volume of the solid made by rotating the region bounded by the given curves around the x-axis.

y = 6x2, y = 0, and x = 1

Homework Equations



V= A(x)*h where h = dx

The Attempt at a Solution



As I was taught in class, I tried using 6x2 as my radius, squaring it, and multiplying it by π, and integrating it (using dx as height of course). The lower limit of integration is 0, as given by the line y=0. The upper limit would need to be the value of y when x=1, which is 6. So, [itex]\int[/itex]36∏x^4 dx with the limits mentioned is what i get for my answer, however I am apparently wrong. What did I do wrong?

EDIT: Never mind, I see the dumb mistake I made in approaching this... it's time for me to go to bed, I've done nothing but study today and I probably need rest. Mods can delete if need be.
 
Last edited:
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  • #2
Well done :)
 

What is "Volume by integration"?

"Volume by integration" is a method used in mathematics and physics to calculate the volume of a three-dimensional object by using calculus. It involves breaking down the object into infinitesimally small pieces and summing up their volumes to find the total volume.

What is the formula for calculating volume by integration?

The formula for calculating volume by integration is ∫A(x)dx, where A(x) represents the cross-sectional area of the object at a given position along the x-axis. This integral is typically evaluated over the entire length of the object, from x=a to x=b.

What types of objects can be measured using volume by integration?

Volume by integration can be used to measure the volume of any three-dimensional object, including simple shapes such as spheres, cylinders, and cones, as well as more complex shapes like irregular solids.

How do you set up the integral for volume by integration?

To set up the integral for volume by integration, you first need to determine the function that represents the cross-sectional area of the object at each position along the axis of rotation. This function is then squared and multiplied by π to find the volume of the infinitesimally small piece. The integral is then evaluated over the entire length of the object to find the total volume.

What are the applications of volume by integration in real life?

Volume by integration has many practical applications, including calculating the volume of containers and tanks, determining the amount of material needed for construction projects, and even measuring the volume of irregularly shaped organs in medical imaging. It is also used in physics to calculate the moment of inertia, which is important in understanding rotational motion.

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