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

Answer 1

#= 29/30 pi#

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 #dA = pi (1 + x + dx)^2 - pi (1 + x)^2#

simplify the algebra with #u = 1+ x# so we have

#dA = pi ((u+dx)^2 - u^2) = pi (2u dx + dx^2)#
#= pi (2 (1+x) dx + dx^2) #

we can already see that #(dA)/dx|_{dx to 0} = pi (2 (1+x) color{red}{+ dx}) #

so we can ignore #mathcal(O)(dx^2)# in the original expression

thus

#dA = 2 pi (1+x) dx #

the volume of that small element is

#dV = (y_2 - y_1) dA#

# = (y_2 - y_1) * 2 pi (1+x) dx #

# = 2 pi (1+x) (sqrt x -x^2) dx #

#\implies V = 2 pi int_0^1 dx qquad (1+x) (sqrt x -x^2) #

#= 2 pi int_0^1 dx qquad (sqrt x -x^2 + x^{3/2} - x^3) #

#= 2 pi (2/3 x^{3/2} -x^3/3 + 2/5x^{5/2} - x^4/4)_0^1 #

#= 2 pi (2/3 - 1/3 +2/5 - 1/4)#

#= 29/30 pi#

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Answer 2

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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Answer from HIX Tutor

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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