How do you use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by #y^2=4x#, x=y revolved about the y-axis?
See below.
Draw a picture (sketch) of the region.
Note that the graphs intersect where
And that happens at
We are asked to use shells, so we take thin representative rectangles parallel to the axis of rotation. In this case that is the
The slice has solid black boundaries and the radius to the axis of rotation is shown as a dashed black line.
Since the thin part is
The lower function is The volume of a representative shell is In this case, we have radius height (details left to the student)
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To find the volume of the solid obtained by rotating the region bounded by (y^2 = 4x) and (x = y) about the y-axis using the method of cylindrical shells, follow these steps:
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Sketch the Region: Draw the curves (y^2 = 4x) and (x = y). Identify the region to be rotated about the y-axis. The intersection points of (y^2 = 4x) and (x = y) give the limits of integration. Solve (y^2 = 4y) to find (y = 0) and (y = 4), which are the limits.
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Set Up the Integral for Volume Using Cylindrical Shells: The formula for the volume (V) of a solid of revolution obtained by rotating a function (f(x)) about the y-axis is: [ V = 2\pi \int_{a}^{b} x \cdot f(x) , dx ] In this case, the cylindrical shell method involves considering a representative shell at a distance (x) from the y-axis, with height (y) given by the curve (y^2 = 4x) or (y = 2\sqrt{x}), and thickness (dx).
However, since one of our boundaries is (x = y), which simplifies as (x), we need to express everything in terms of (y) because our rotation is around the y-axis and our bounds are in terms of (y). The region is between (y=0) and (y=4), and the original equation (y^2 = 4x) can be rearranged as (x = \frac{y^2}{4}).
For a shell at (y), the height is the difference in (x)-values across the shell, which is simply (x = \frac{y^2}{4}) since the left boundary is at (x=0) (the y-axis), and the right boundary is described by (x = \frac{y^2}{4}). The radius of each shell is (y).
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Compute the Integral: The volume integral becomes: [ V = 2\pi \int_{0}^{4} y \left(\frac{y^2}{4}\right) dy = 2\pi \int_{0}^{4} \frac{y^3}{4} dy ]
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Evaluate the Integral: Solve the integral: [ V = 2\pi \left[\frac{1}{4} \cdot \frac{y^4}{4}\right]_0^4 = \frac{\pi}{8} [y^4]_0^4 = \frac{\pi}{8} \cdot (4^4 - 0^4) = \frac{\pi}{8} \cdot 256 = 32\pi ]
Thus, the volume of the solid obtained by rotating the given region about the y-axis is (32\pi) cubic units.
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To use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by ( y^2 = 4x ) and ( x = y ) revolved about the y-axis:
- Express the given equations in terms of ( y ) to determine the limits of integration.
- Set up the integral for the volume using the formula for cylindrical shells.
- Integrate the expression with respect to ( y ) over the determined limits to find the volume.
The cylindrical shell method involves summing the volumes of infinitely thin cylindrical shells that approximate the solid of revolution. Each cylindrical shell has a radius determined by the distance from the axis of rotation and a height determined by the function describing the shape being rotated. Integrating these shells over the appropriate range provides the volume of the solid.
In this case, since the region is revolved about the y-axis, the radius of each cylindrical shell will be ( y ), and the height will be the difference between the two functions ( \sqrt{4y} - y ).
After setting up the integral with the appropriate limits, integrate with respect to ( y ) to find the volume of the solid of revolution.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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