# How do you find the volume bounded by #y=x^2#, #x=y^2# revolved about the x=-1?

Consider the small element width dx as shown, being revolved around the line x = -1

The cross sectional area (csa) of the washer of width dx is the csa of the outer circle minus the csa of the inner

ie

simplify the algebra with

we can already see that

so we can ignore

thus

the volume of that small element is

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To find the volume bounded by the curves (y=x^2) and (x=y^2) revolved about the line (x=-1), you can use the method of cylindrical shells. First, you need to find the points of intersection between the two curves by setting (y=x^2) equal to (x=y^2) and solving for (x) and (y). The points of intersection are ((-1, 1)) and ((0, 0)). Then, you integrate from (y=0) to (y=1) using the formula for the volume of cylindrical shells:

[V = \int_{0}^{1} 2\pi \cdot \text{radius} \cdot \text{height} , dy]

The radius is the distance from the line of revolution ((x=-1)) to the curve, which is (1+y). The height is the difference between the (x)-values of the two curves, which is (\sqrt{y} - (-1)). Thus, the integral becomes:

[V = \int_{0}^{1} 2\pi \cdot (1+y) \cdot (\sqrt{y} + 1) , dy]

Solving this integral will give you 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.

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