How do you find the area of the surface generated by rotating the curve about the yaxis #y=x^2, 0<=x<=2#?
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To find the area of the surface generated by rotating the curve ( y = x^2 ) about the yaxis 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 yaxis 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.

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

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 yaxis from ( x = 0 ) to ( x = 2 ).
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When evaluating a onesided 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 onesided 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 onesided 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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