# Is this the right integral set-up to find the volume?

• Lo.Lee.Ta.
In summary, the conversation discusses finding the volume between two functions when revolved around a specific axis using the washer method. It also addresses the proper setup for the inner and outer radii in this scenario.
Lo.Lee.Ta.
1. Find volume between f(x) = x, g(x)= sin(sqrt(5x+3)), x=1, and x=2, when revolved around y=4.

2. Would it be correct to write the integral like this?

∫1 to 2 of [$\pi$(4 - sin(√(5x + 3))2 - $\pi$(4 - x)2]

I am using the washer method, and for the gap that's in the middle I usually think about it by saying r= inner curve - axis of rotation.
R= outer curve - axis of rotation

But in this case the axis of rotation is above the function, so would it be r= 4 - inner curve

and R= 4 - outer curve?

Thanks! :)

Lo.Lee.Ta. said:
∫1 to 2 of [$\pi$(4 - sin(√(5x + 3))2 - $\pi$(4 - x)2]
Yes, that's right. You ought to check that the curves do not cross within the range.

Yes, the inner radius should be 4 - inner curve and the outer radius should be 4 - outer curve. I think your setup is correct.

:D Thanks so much, haruspex and JPaquim! :D

## 1. How do I know if the integral set-up is correct for finding the volume?

The integral set-up is correct for finding the volume if the integrand is a function of the variable of integration and the limits of integration are appropriate for the given volume. Additionally, the integral should be set up in the correct orientation (e.g. horizontal or vertical) depending on the shape of the volume.

## 2. Can the integral set-up be used for any type of volume?

The integral set-up can be used for a variety of volumes, including rectangular prisms, cylinders, cones, and pyramids. However, for more complex shapes, other methods such as triple integrals may be necessary.

## 3. Do I need to use a specific integration technique for finding the volume?

The integration technique used depends on the shape of the volume and the given integrand. For simple shapes, the volume can be found using basic integration techniques such as the power rule or substitution. For more complex shapes, techniques such as integration by parts or partial fractions may be necessary.

## 4. Is there a certain order in which the variables of integration should be chosen?

The order of the variables of integration can affect the set-up of the integral, but ultimately, the correct limits of integration should still be chosen. It is important to choose the order that makes the integral easier to evaluate.

## 5. What should I do if I am unsure about the set-up for finding the volume?

If you are unsure about the set-up for finding the volume, it is best to consult with a professor or colleague who is knowledgeable in the subject. You can also refer to textbooks or online resources for examples and guidance on setting up integrals for different types of volumes.

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