Finding volume using integrals

In summary, the conversation is about finding the volume of a solid generated by revolving a region bounded by a curve and a line about the x-axis. The solution involves using an integral and recognizing the symmetry of the function, resulting in a volume of (128 pi)/3. However, it is noted that the solution should be multiplied by 2 since the function is defined from -4 to 4. It is also mentioned that the figure formed is a sphere of radius 4.
  • #1
donjt81
71
0
So here is the question

find the volume of the solid generated by revolving the region bounded by the curve y = sqrt(16 - x^2) and the line y = 0 about the x axis.

this is how I solved it

[tex] \int_{0}^{4} \Pi (16 - y^{2}) \; dy [/tex]
[tex] \Pi \int_{0}^{4} (16 - y^{2}) \; dy [/tex]

pi(16*y - y^3/3) from 0 to 4
pi(16*4 - 64/3 - 0)

so the answer I got is (128 pi)/3

is this correct.
 
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  • #2
Unless I'm missing something, you should multiply your whole integral with 2, since your function is defined from -4 to 4 and symmetric.
 
  • #3
Well, why haven't you used -4 as your lower limit?
Note that this would double the volume of your object.
 
  • #4
Another check is to recognize that [itex]y= \sqrt{16- x^2}[/itex] is the upper half of a circle and so the figure formed is a sphere of radius 4. Its volume is [itex](4/3)\pi (4)^3[/itex], twice your answer.
 
  • #5
thanks guys i completely missed that. you are right it should be -4 to 4... other than that does everything else look ok.
 

Related to Finding volume using integrals

What is volume and how is it related to integrals?

Volume is a measure of the amount of space occupied by an object. In mathematics, it is often calculated using integrals, which are mathematical tools used to find the area under a curve. In the context of finding volume, integrals are used to find the area of a cross-section at each point along the length of an object, and then these areas are added together to find the total volume.

What is the formula for finding volume using integrals?

The formula for finding volume using integrals is V = ∫A(x)dx, where V is the volume, A(x) is the area of the cross-section at a given point, and dx represents an infinitesimally small change in the variable x. This formula is based on the fact that volume can be thought of as the sum of infinitely small slices or cross-sections of an object.

What are the steps involved in finding volume using integrals?

The steps involved in finding volume using integrals are:

  1. Identify the shape of the object for which you want to find the volume.
  2. Choose a variable (usually x) to represent the length of the object.
  3. Find the formula for the area of the cross-section at each point along the length of the object.
  4. Set up the integral by plugging in the formula for the area and integrating with respect to the chosen variable.
  5. Use the limits of integration to determine the starting and ending points of the object.
  6. Evaluate the integral to find the total volume.

What are some real-world applications of finding volume using integrals?

Finding volume using integrals has many real-world applications, including calculating the volume of irregularly shaped objects, such as rocks or trees, in geology and forestry. It is also used in engineering to determine the volume of materials needed for construction projects. In physics, it can be used to calculate the volume of a fluid in a container or the volume of an object submerged in water.

What are some common challenges when using integrals to find volume?

Some common challenges when using integrals to find volume include:

  • Choosing the correct variable to represent the length of the object.
  • Identifying the correct formula for the area of the cross-section at each point.
  • Determining the limits of integration, which can be tricky for complex or irregularly shaped objects.
  • Evaluating the integral, which may require advanced mathematical techniques.
  • Dealing with units of measurement and ensuring they are consistent throughout the calculation.

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