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Calculate the length of the curve r(t) = (t, (1-t^2)^1/2) t ∈ (-1,1)

User Ivica
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Final answer:

To calculate the length of the curve r(t) = (t, (1-t^2)^1/2) on the interval [-1, 1], we can use the arc length formula. By finding the derivative of the curve and substituting it into the arc length formula, we can calculate the length of the curve as 4/3.

Step-by-step explanation:

To calculate the length of the curve, we can use the arc length formula. The arc length of a curve y = f(x) on the interval [a, b] is given by:

L = ∫(a to b) √(1 + (f'(x))^2) dx

In this case, the curve is defined parametrically as r(t) = (t, (1 - t^2)^(1/2)). To find the length of the curve, we need to find the derivative of r(t) and substitute it into the arc length formula:

L = ∫(-1 to 1) √(1 + (-2t)^2) dt

Simplifying and evaluating the integral, we get L = 4/3.

User Bastl
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