Cloudless said:Homework Statement
The given curve is r(t) = <t2, 2t, -3>
Write an equation for the length of the curve from <0,0,-3> to <1, 2, -3>
2. The attempt at a solution
I take the derivative of r(t) for r'(t), then plug it into the length formula.
L = ∫ of √( (2t)2 + 22 )
For the Integral, I put from 0 to 1, because in the original equation, the x component t2 = 1 when t is 1, and the y component 2t = 2 when t is 1.
Am I doing something wrong? o_o
The equation for the length of a curve is the integral of the square root of the sum of the squares of the derivatives of the curve with respect to the independent variable.
The integral ends represent the interval over which the curve is being measured. This interval can be specified by providing the lower and upper limits of the independent variable.
To calculate the length of a curve using the integral ends, you first need to determine the curve's equation and its derivatives. Then, plug these values into the length equation and evaluate the integral using the provided limits.
Yes, the integral ends can be negative or non-numeric values as long as they are within the interval of the curve being measured. However, it is important to ensure that the integral ends are within the domain of the curve's equation.
Yes, there are some limitations to using the equation for length of a curve. This equation assumes that the curve is continuous and smooth, and the integral ends must be within the domain of the curve's equation. Additionally, it may not be possible to calculate the length of a curve with complex or non-analytic equations.