Calculate the area of the region between the graph of the function f=(x-1)*x^(-2), x axis and the lines x=1 and x=e?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To calculate the area of the region bounded by the graph of the function f, x axis and the given lines x=1 and x=e, we must to evaluate the definite integral of the function f(x) = (x-1)/x^2

Int f(x)dx=Int [(x-1)/x^2]dx

We'll aply the property of integrals to be additive:

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

Int (1/x)dx - Int [x^(-2)]dx = ln x - [x^(-2+1)]/(-2+1)

We'll apply Leibniz-Newton formula:

Int f(x)dx= F(e) - F(1)

F(e) - F(1) = (ln e - ln 1) + (1/e - 1)=

= 1-0+(1/e)-1=1/e

The value of the area of the region bounded by the graph of the function f, x axis and the given lines x=1 and x=e is A = 1/e square units.

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