How do you find the area of the surface generated by rotating the curve about the y-axis #y=x^2, 0<=x<=2#?

Answer 1

#pi/6(17sqrt17-1)#

Since we are rotating this solid around the #y#-axis, we are concerned with the #x# distance from the #y#-axis to the function. This relation is given by #x=pmsqrty#. We're only dealing with positive #x# values, so we can reduce this to just #x=sqrty# for our case.
The formula for the surface area of a solid generated by rotating some curve #g(y)# around the #y#-axis on #yin[c,d]# is given by
#A=2piint_c^dg(y)sqrt(1+(g'(y))^2)dy#
We go from #x=0# to #x=2#, which is analogous to traveling from #y=0# to #y=4#, which is what we care about.
We will use #g(y)=sqrty#. Note that #g'(y)=1/(2sqrty)#.
#A=2piint_0^4sqrtysqrt(1+(1/(2sqrty))^2)dy#
#color(white)A=2piint_0^4sqrt(y(1+1/(4y)))dy#
#color(white)A=2piint_0^4sqrt(y+1/4)dy#
Let #u=y+1/4#. This implies that #du=dy#. We will also have to change the bounds.
#A=2piint_(1//4)^(17//4)u^(1/2)du#
#color(white)A=2pi[2/3u^(3/2)]_(1//4)^(17//4)#
#color(white)A=(4pi)/3((17/4)^(3/2)-(1/4)^(3/2))#
Note that #4^(3/2)=8#:
#A=(4pi)/3((17^(3/2)-1)/8)#
#color(white)A=pi/6(17sqrt17-1)#
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Answer 2

To find the area of the surface generated by rotating the curve ( y = x^2 ) about the y-axis from ( x = 0 ) to ( x = 2 ), you can use the formula for the surface area of a solid of revolution.

The formula to find the surface area ( A ) of the surface generated by rotating a curve ( y = f(x) ) about the y-axis from ( x = a ) to ( x = b ) is given by:

[ A = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]

For the curve ( y = x^2 ) from ( x = 0 ) to ( x = 2 ), we need to find ( \frac{dy}{dx} ) and then apply the formula.

  1. Find ( \frac{dy}{dx} ) for ( y = x^2 ): [ \frac{dy}{dx} = \frac{d}{dx}(x^2) = 2x ]

  2. Substitute ( \frac{dy}{dx} = 2x ) into the formula: [ A = 2\pi \int_{0}^{2} x^2 \sqrt{1 + (2x)^2} , dx ]

Now, you can integrate this expression to find the surface area. Once you've evaluated the integral, you'll have the surface area of the solid of revolution generated by rotating the curve ( y = x^2 ) about the y-axis from ( x = 0 ) to ( x = 2 ).

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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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