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: a₁, a₂, … ,a_k / f(a_i) = 0 and then:

Example: calculate the area among the curve y = x³ – x, the abscissa axis, and the straight lines x = 0 and x = 2

The 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 a₁, a₂, … ,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) = x², g(x) = -2x² + 3

Example 2: calculate the area between the graphs of the functions: f(x) = x³, g(x) = 4x

Exercises:

1.- Calculate the area among the graph of the function f(x) = x² and the straight lines x = 1 and x = 3

2.- (PAEG- June 2012) Calculate the area enclosed by the parabola f(x) = x² - 1 and the straight line g(x) = 5 - x

3.- (PAEG- September 2012) Calculate the area enclosed by the graphs of the functions f(x) = x³ - 3x² + 2x + 1 and g(x) = 1

Solutions:

A = 26/3 u²

A = 125/6 u²

A = 1/2 u²