How to find the radius in each of these integrals?

In summary, to find the radius in each of these integrals, one must first identify the type of integral being solved (e.g. indefinite or definite). Then, the appropriate integration techniques such as substitution, integration by parts, or trigonometric identities can be used to solve for the radius. It is also important to carefully consider the given boundaries or limits of integration in order to accurately determine the radius. With practice and understanding of integration principles, finding the radius in integrals can become a straightforward process.
  • #1
Celestor
1
0

Homework Statement



Given: y = sqrt(x), y=0, x=3

Find the volume of the solid bounded by these functions, revolved around B) y-axis and C) line x=3. (Disk method)

Homework Equations


Disk method of finding volume using π ∫ r2dy

The Attempt at a Solution



Ok so, the part of the problem that I am having trouble on is how to find the radius. I know that both integrals will be with respect to dy with the limits being 0 and sqrt(3) and I understand that these two integrals SHOULD be different, but I'm not sure what that difference is in terms of the radius in πr2.

Please refer to the solution to B and C given in this link (http://www.calcchat.com/book/Calculus-10e/7/2/11/). My question is, why is the radius [32 - (y2)2] in B and why is the radius [3-y2]2 in C. My problem is understanding why the power of 2 is distributed differently and how I'm supposed to interpret that based on the problem.

Thanks,
 
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  • #2
Hello Celestor, welcome to PF :smile: !

Your question is a bit strange. The radius itself isn't [ 32-(y2)2 ] in b. What do you get when you rotate the shaded area around the y-axis ?

[edit] I mean the dark shaded rectangular area...​

In (c) you rotate around the line x = 3, so what do you get when you rotate the shaded area around the dashed line ?
[edit] cute that you get to see the exercise worked out at the website. But doesn't that prevent you from first trying to find your own path towards a solution ?​
 
Last edited:

FAQ: How to find the radius in each of these integrals?

How do I identify the radius in an integral?

The radius is typically the distance from the center of a circle or sphere to the outer edge. In integrals, it is often represented by the letter r or a variable such as x or y.

Can I find the radius by looking at the integrand?

Yes, in some cases, the integrand (the function inside the integral) may include the radius as a variable. For example, if you see an expression like √(r^2 - x^2), the radius would be represented by r.

How can I find the radius if it is not explicitly given in the integral?

If the radius is not explicitly given in the integral, you can often find it by looking at the boundaries of integration. For example, if you are integrating with respect to x and the boundaries are from 0 to r, then r would be the radius.

What if the integral involves a three-dimensional shape, like a sphere?

In three-dimensional integrals, the radius is typically represented by either r or ρ (rho). Similar to two-dimensional integrals, you can often find the radius by looking at the boundaries of integration.

Is it always necessary to find the radius in an integral?

No, not all integrals require the radius to be explicitly identified. It depends on the specific problem and what information is given. Sometimes, the radius may be already known or can be determined from other given variables.

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