Volume by Rotating a Curve: Finding the Solid Between Two Curves

In summary, the conversation discusses a problem involving finding the volume of a solid obtained by rotating a region bounded by two curves about the x-axis. The person attempted to use cylindrical shells and integrated using an incorrect formula. They were advised to use the correct formula and check their upper and lower limits, which they confirmed were correct.
  • #1
mikhailpavel
16
0

Homework Statement


Hey i have a problem here with volume by cylindrical shells. i wanted to find the given volume of the solid obtained by rotating the region bounded by the curves x=1+(y-2)^2 and x=2 about the x axis.


Homework Equations


we tried to integrating 2 phi f(y) dy with upper limit 3 and lower limit 1.


The Attempt at a Solution


we got the answer 33.5 but i don't think it is the correct one.
immediate help will be appreciable.
 
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  • #2
You are using cylindrical shells, but your formula is incorrect. A typical shell in this problem has a volume of 2pi*y*(x2 - x1)*dy

x2 is the x-value on the line x = 2, and x1 is the x-value on the parabola. You have graphed the region being revolved, right?
 
  • #3
can u tell me if my upper and lower limits are right because still i am not getting the right answer..i think!
 
  • #4
Yes, the two curves intersect at (2, 1) and (2, 3) so y ranges between 1 and 3.
 

FAQ: Volume by Rotating a Curve: Finding the Solid Between Two Curves

What is the concept of volume by rotating a curve?

The concept of volume by rotating a curve is a mathematical method used to calculate the volume of a three-dimensional shape formed by rotating a two-dimensional curve around an axis. This method is commonly used in calculus and geometry to find the volume of objects such as cylinders, cones, and spheres.

How is the volume calculated using this method?

The volume by rotating a curve is calculated by using the formula V = πr^2h, where r is the radius of the curve and h is the height or length of the curve. This formula is based on the fact that the volume of a cylinder is equal to the area of its base (formed by the curve) multiplied by its height.

What types of curves can be used to calculate volume by rotation?

Any continuous curve can be used to calculate volume by rotation, as long as it is rotated around a fixed axis. This includes curves such as circles, ellipses, parabolas, and more complex curves like spirals and helixes.

What are some real-world applications of volume by rotating a curve?

Volume by rotating a curve has many practical applications, including determining the volume of containers such as bottles and cans, calculating the volume of pipes and tubes, and finding the volume of objects like gears and water wheels.

Are there any limitations to using this method for calculating volume?

While volume by rotating a curve is a useful method, it does have limitations. This method can only be used to find the volume of objects with a circular cross-section, and it cannot be applied to irregular or non-continuous shapes. Additionally, this method assumes that the curve being rotated is a perfect shape with a constant radius, which may not always be the case in real-world scenarios.

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