DefiniteIntegralApplications

# calculation of areas

- If f is positive:

- If f is negative:

- If f is positive and negative, we have to find out its zeros in the interval: a_{1}, a_{2}, … ,a_{k} / f(a_{i}) = 0 and then:

Example: calculate the area among the curve y = x

^{3}– x, the abscissa axis, and the straight lines x = 0 and x = 2The area between the graphs of two functions, f and g, in [a,b]:

–If f ≤ g or f ≥ g in the interval

–Otherwise, we find out a_{1}, a_{2}, … ,a_{k} in [a,b] / f(a_{i}) = g(a_{i}) and then:

Example 1: calculate the area between the graphs of the functions: f(x) = x

^{2}, g(x) = -2x^{2}+ 3Example 2: calculate the area between the graphs of the functions: f(x) = x

^{3}, g(x) = 4x**Exercises**:

1.- Calculate the area among the graph of the function f(x) = x

^{2}and the straight lines x = 1 and x = 32.- (PAEG- June 2012) Calculate the area enclosed by the parabola f(x) = x

^{2}- 1 and the straight line g(x) = 5 - x3.- (PAEG- September 2012) Calculate the area enclosed by the graphs of the functions f(x) = x

^{3}- 3x^{2}+ 2x + 1 and g(x) = 1Solutions:

1.

A = 26/3 u

^{2}2.

A = 125/6 u

^{2}3.

A = 1/2 u

^{2}

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