# How do you find the volume of the solid generated by revolving the region enclosed by the parabola #y^2=4x# and the line y=x revolved about the y-axis?

The volume of the solid generated by revolving the who region bounded by the given parabola and the straight line would thus be

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To find the volume of the solid generated by revolving the region enclosed by the parabola (y^2=4x) and the line (y=x) revolved about the y-axis, you can use the method of cylindrical shells. The volume can be calculated using the formula:

[V = 2\pi \int_{a}^{b} x \cdot f(x) , dx]

where (f(x)) represents the height of the shell at a given x-value, and (a) and (b) are the x-values at the points of intersection of the parabola and the line.

First, find the intersection points of the parabola and the line by substituting (y=x) into the equation of the parabola (y^2=4x). This gives (x^2=4x), which simplifies to (x(x-4)=0), so (x=0) and (x=4).

Now, the height of the shell is the difference between the y-values of the parabola and the line, which is (f(x) = y_{\text{parabola}} - y_{\text{line}}). Substituting the equations of the parabola and the line, we get (f(x) = \sqrt{4x} - x).

Finally, integrate (x \cdot f(x)) from (x=0) to (x=4) and multiply the result by (2\pi) to find the volume of the solid. This integration will yield the volume of the solid generated by revolving the region about the y-axis.

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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.

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