klepkowy7c

2022-07-20

How do I know which method of revolution to use for finding volume in Calculus?
Is there any easy way to decide which method I should use to find the volume of a revolution in Calculus? I'm currently in the middle of my second attempt at Calculus II, and I am getting tripped up once again by this concept, which seems like it should be rather straight forward, but I can't figure it out. If I have the option of the disk/washer method and the shell method, how do I know which one I should use?

Coleman Ali

Expert

Step 1
I think that there are some standard formulae in integral calculus to calculate the Volume of revolution. These offer you a straight forward method ( I don't know anything about the dish washer method ) but you still have to know how to trace the curves to apply the following formulae on them:-
$V=\pi {\int }_{a}^{b}{y}^{2}dx$ where $y=f\left(x\right)$ is the equation of the curve and $x=a$ and $x=b$ are the limits of the revolution.
Step 2
$V=\pi {\int }_{c}^{d}{x}^{2}dy$ where $x=g\left(y\right)$ is the equation of the curve and $y=c$ and $y=d$ are the limits of the revolution.
To calculate the volume of revolution when polar coordinated are given, you substitute $y=cos\theta$ and $x=sin\theta$ and calculate the limits accordingly. Also, you need to see that the given formulae are valid only when the revolution is occuring about the x-axis or y-axis.If the curve is revolving around $x=$ constant or $y=$ constant, then you need to change the origin and make suitable adjustements.

Paxton Hoffman

Expert

Step 1
The first thing to understand is that you don’t directly choose the method of integration: you determine what kind of integration will be easier, based on the shape of the region in question, and that determines which method you’ll use.
Draw the region that’s being revolved. Then ask yourself: does it slice up nicely into vertical strips, or do horizontal strips work better?
1. If the region has boundaries of the form $y=f\left(x\right)$, $y=g\left(x\right)$, $x=a$, and $x=b$, the answer is almost always that vertical strips are simpler: for each x from a to b you have a strip of length $f\left(x\right)-g\left(x\right)$ or $g\left(x\right)-f\left(x\right)$, depending on which of f(x) and g(x) is larger.
2. Similarly, if the region has boundaries of the form $x=f\left(y\right)$, $x=g\left(y\right)$, $y=a$, and $y=b$, the answer is almost always that horizontal strips are simpler: for each y from a to b you have a strip of length $f\left(y\right)-g\left(y\right)$ or $g\left(y\right)-f\left(y\right)$, depending on which of f(y) and g(y) is larger.
3. The case of a region bounded by $y=f\left(x\right)$ and $y=g\left(x\right)$, where you have to solve for the points of intersection of the two curves in order to find the horizontal extent of the region, is really just special case of (1). Similarly, if the region is bounded by $x=f\left(y\right)$ and $x=g\left(y\right)$, and you have to solve for the vertical extent of the region, you’re looking at a special case of (2).
What you want is a way of slicing the region into vertical or horizontal strips whose endpoints are defined in the same way. Take, for instance, the region bounded by $y=x$ and $y=x\left(x-1\right)$. If you slice it into vertical strips, each strip runs from the parabola at the bottom to the straight line at the top, so the strip at each x between 0 and 2 has its bottom end at $x\left(x-1\right)$ and its top end at x. If you were to slice it into horizontal strips, the ones between $y=0$ and $y=2$ would have their left ends on the straight line and their right ends on the parabola, but the ones between $y=-1/4$ and $y=0$ would have both ends on the parabola. Thus, you’d need a different calculation to handle the part of the region below the x-axis from the one that you’d need for the part above the x-axis.
Whether the boundaries are given in the form $y=f\left(x\right)$ or the form $x=f\left(y\right)$ is often a good indication: the former tends to go with vertical slices and the latter with horizontal slices. It’s far from infallible, however, and in some problems some boundaries are given in one form and some in the other. You should always look at a picture of the region. You want slices whose endpoints are defined consistently, and you want slices that don’t have any gaps in them.
Once you’ve decided which way to slice up the region, sketch in the axis of revolution. If it’s parallel to your slices, each slice will trace out a cylindrical shell as it revolves about the axis. If, on the other hand, it’s perpendicular to your slices, each slice will trace out a washer or disk as it revolves about the axis. In either case the proper method of integration has automatically been determined for you.

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