whatlifeforme said:
whatlifeforme said:
tiny-tim said:ie, what is √(4x4 + 4x2 + 1) ?
The formula for finding the length of a curve is given by the arc length formula, which is ∫√(1+(dy/dx)^2) dx, where dy/dx represents the derivative of the function.
You can find the derivative of a curve by using the power rule, chain rule, or product rule, depending on the complexity of the function. In this case, the derivative of y=(2/3)(x^2+1)^(3/2) would be (4/9)x(x^2+1)^(1/2).
No, this formula can only be used for finding the length of a smooth curve, meaning it has a continuous first derivative.
To find the length of the curve y=(2/3)(x^2+1)^(3/2), you need to first calculate the derivative, then plug it into the arc length formula: ∫√(1+(dy/dx)^2) dx. Next, you need to determine the limits of integration, which would be the points where the curve starts and ends. Finally, you can evaluate the integral to find the length of the curve.
Yes, this formula can be used for both parametric and polar curves with some modifications. For parametric curves, the formula would be ∫√(x'(t)^2+y'(t)^2) dt. For polar curves, the formula would be ∫√(r^2+(dr/dθ)^2) dθ.