Yes. You need to figure out the initial ##t_1## final value ##t_2## of ##t##, then do the integralTurbodog66 said:Homework Statement
Consider the path r(t) = <10t,5t2,5ln(t) defined for t >0. Find the length of the curve between (10,5,0) and (20,20,5ln(2))
Homework Equations
L= ∫ab |r'(t)|dt
The Attempt at a Solution
r'(t) = <10, 10t, 5/t>
t values are 1 and 2 based on the x values for the points listed
√(102 + (10t)2 + (5/t)2)
√(100 + 100t2 + 25/t2)
√(25) ⋅ √(4 + 4t2 + 1/t2)
5√(4 + 4t2 + 1/t2)This is where I am getting stuck. Is there an additional step that is missing?
Turbodog66 said:Homework Statement
Consider the path r(t) = <10t,5t2,5ln(t) defined for t >0. Find the length of the curve between (10,5,0) and (20,20,5ln(2))
Homework Equations
L= ∫ab |r'(t)|dt
The Attempt at a Solution
r'(t) = <10, 10t, 5/t>
t values are 1 and 2 based on the x values for the points listed
√(102 + (10t)2 + (5/t)2)
√(100 + 100t2 + 25/t2)
√(25) ⋅ √(4 + 4t2 + 1/t2)
5√(4 + 4t2 + 1/t2)This is where I am getting stuck. Is there an additional step that is missing?
Ray Vickson said:Yes. You need to figure out the initial ##t_1## final value ##t_2## of ##t##, then do the integral
$$\text{Length} = \int_{t_1}^{t_2} 5 \sqrt{ \displaystyle 4 + 4 t^2 + \frac{1}{t^2}} \, dt $$
Turbodog66 said:Based on the two points listed, I found that t1 = 1 and t2=2. So next I would then substitute in the t values and solve? I attempted that previously, but I did not get the correct answer.
$$\text{Length} = \int_{1}^{2} 5 \sqrt{ \displaystyle 4 + 4 (2^2 -1^2) + \frac{1}{2^2-1^2}} \, dt $$
The formula for calculating the length of a curve between two points is given by the arc length formula: L = ∫ab √(1 + [f'(x)]2) dx, where a and b are the starting and ending points of the curve and f(x) is the function representing the curve.
The arc length formula is derived using the Pythagorean theorem and the concept of infinitesimal intervals. By dividing the curve into small intervals and approximating each interval with a straight line, we can use the Pythagorean theorem to calculate the length of each interval. By taking the limit of these intervals as they tend to zero, we can derive the arc length formula.
Yes, the arc length formula can be applied to any curve as long as it can be represented by a continuous function. This includes curves that are not necessarily smooth, such as piecewise functions.
The parameterization of a curve can affect the calculation of its length because it determines the direction in which the curve is traversed. A change in the parameterization may result in a different length being calculated for the same curve.
Yes, the arc length formula can be extended to higher dimensions, such as curves in three-dimensional space. The formula becomes L = ∫ab √(1 + [f'(t)]2 + [g'(t)]2) dt, where f(t) and g(t) represent the parametric equations for the curve in the x and y directions, respectively.