Expert: Ahmed Salami - 8/30/2010

Question
please find the area between the graphs of the functions given by y= -4x+3 and y=x^2+6

Answer
Hi Steve,
The functions intercept at
-4x + 3 = x² + 6
x² + 4x + 3 = 0
(x + 3)(x + 1) = 0
x = -1 or -3
The area between the graphs is the difference between the areas they make with the x-axis.
A = ∫(-4x + 3)dx - ∫(x² + 6)dx        (between the limits -3 to -1)
 = ∫[(-4x + 3) - (x² + 6)]dx
 = ∫(-4x - x² - 3)]dx
 = [-2x² - x³/3 - 3x]     (between the limits -3 to -1)
 = [-2(-1)² - (-1)³/3 - 3(-1)] - [-2(-3)² - (-3)³/3 - 3(-3)]
 = [-2 + 1/3 + 3] - [-18 + 9 + 9]
 = (1 + 1/3) - 0
 = 4/3 square units

Regards