Theory question 110
Integration(a) Using the same axes, sketch the curves y = 6 − x − x2 and y = 3x2 − 2x + 3.
(b) Find the x-coordinates of the points of intercession of the two curves in (a)
(c) Calculate the area of the finite region bounded by the two curves (a).
Show the model answer
Model answer
(a) y = 6 − x − x2: ∩-shaped, cuts x-axis at −3 and 2, y-intercept 6, maximum at (−½, 6¼)
y = 3x2 − 2x + 3: ∪-shaped, y-intercept 3, minimum at (⅓, 2⅔), does not cut the x-axis
(b) 6 − x − x2 = 3x2 − 2x + 3 → 4x2 − x − 3 = 0 → (4x + 3)(x − 1) = 0
x = −3/4 or x = 1
(c) Area = ∫ from −3/4 to 1 of (3 + x − 4x2)dx = [3x + x2/2 − 4x3/3]
= 13/6 − (−45/32) = 343/96 ≈ 3.57 square units
Explanation
Area between curves = ∫(upper − lower)dx between the intersection points.