What is the surface area of the solid created by revolving #f(x) = x^2-3x+2 , x in [3,4]# around the x axis?
The surface area of a solid created by revolving
#SA = 2pi \ int_(x=alpha)^(x=beta) \ f(x) \ sqrt(1+(f'(x))^2) \ dx#
So we have;
# \ \ \ \ \ \ f(x) = x^2 - 3x + 2 #
# :. f'(x) = 2x-3 #
And so the Surface Area is;
# SA = 2pi \ int_(3)^(4) \ (x^2-3x+2) \ sqrt(1+(2x-3)^2) \ dx #
#\ \ \ \ \ = 2pi \ int_(3)^(4) \ (x^2-3x+2) \ sqrt(1+(4x^2-12x+9)) \ dx#
#\ \ \ \ \ = 2pi \ int_(3)^(4) \ (x^2-3x+2) \ sqrt( 4x^2-12x+10 ) \ dx# Although the integral can be established it is quite complex,so I will just quote the result
# SA = pi/32 (5sinh^-1(3)-45sqrt(10) -5sinh^-1(5) +235sqrt(26))#
#\ \ \ \ \ = 103.426839 ... # #
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To find the surface area of the solid created by revolving the function ( f(x) = x^2 - 3x + 2 ) over the interval ( x \in [3,4] ) around the x-axis, you can use the formula for the surface area of a solid of revolution:
[ S = \int_{a}^{b} 2\pi \cdot f(x) \cdot \sqrt{1 + \left(\frac{dy}{dx}\right)^2} , dx ]
where ( a ) and ( b ) are the limits of integration (in this case, 3 and 4), ( f(x) ) is the given function, and ( \frac{dy}{dx} ) represents the derivative of the function with respect to ( x ).
First, find the derivative of ( f(x) ):
[ f'(x) = \frac{d}{dx}(x^2 - 3x + 2) = 2x - 3 ]
Next, plug in the values into the formula and evaluate the integral:
[ S = \int_{3}^{4} 2\pi \cdot (x^2 - 3x + 2) \cdot \sqrt{1 + (2x - 3)^2} , dx ]
After solving the integral, you will get the surface area 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.
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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