Graphs.Find the length of the graph of f(x) = (1-x^2)^1/2 on [0, b].

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll use the formula of calculating the length of the graph:

L = Int sqrt{1+[f'(x)]^2} from x = a to x = b

In our case a = 0 and b = b

The length of the graph of f(x) = sqrt(1-x^2) is:

L = Int sqrt{1+[(sqrt(1-x^2))']^2}dx , from 0 to b.

We'll calculate the derivative of sqrt(1-x^2)

[sqrt(1-x^2)]' = -2x/2sqrt(1-x^2)

[sqrt(1-x^2)]' = -x/sqrt(1-x^2)

{[sqrt(1-x^2)]'}^2 = x^2/(1-x^2)

We'll add 1 both sides:

1 + {[sqrt(1-x^2)]'}^2 = 1 + x^2/(1-x^2)

1 + {[sqrt(1-x^2)]'}^2 = (1 - x^2 + x^2)/(1-x^2)

We'll eliminate like terms:

1 + {[sqrt(1-x^2)]'}^2 = 1/(1-x^2)

L = Int dx/(1-x^2)

L = arcsin x from x = 0 to x = b

L = arcsin b - arcsin 0

L = arcsin b

The length of the graph of the function f(x) = sqrt(1-x^2), over the interval [0,b], is L = arcsin b.

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