# Find the exact length of the curve?

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#y=x^3/3+1/(4x)# , #1\lex\le2#

Factor by symmetry:

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To find the exact length of a curve, you would typically use calculus and integrate the square root of the sum of the squares of the derivatives of the curve with respect to the independent variable. This process is known as arc length integration. The formula for finding the arc length ( L ) of a curve ( y = f(x) ) from ( x = a ) to ( x = b ) is:

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

Where ( \frac{dy}{dx} ) represents the derivative of ( y ) with respect to ( x ). You integrate this expression over the given interval from ( a ) to ( b ). Once you perform the integration, you'll have the exact length of the curve between the specified points.

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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.

- Given #x^t * y^m - (x+y)^(m+t)=0# determine #dy/dx# ?
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- If #x^2 y=a cos#x, where #a# is a constant, show that #x^2 (d^2 y)/(dx^2 )+4x dy/d +(x^2+2)y=0 #?
- What is the arclength of #f(x)=e^(1/x)/x-e^(1/x^2)/x^2+e^(1/x^3)/x^3# on #x in [1,2]#?
- What is the general solution of the differential equation #(1+x^2)dy/dx + xy = x #?

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