Subject: Graphing regions

Name: Esther

Who is asking: Parent
Level: Secondary

Question: This is the second of two questions my son had problems with.

Find the area of the region enclosed by the graphs of y = x3-6x and y = -2x between their points of intersection.

Hi Esther,

Let f(x) = x3-6x and g(x) = -2x. First you need to find where the curves intersect. Solving f(x) = g(x) gives

x3 - 6x = -2x
x(x2 - 4) = 0
x = -2, 0 or 2 Thus the curves cross at three points, where x = -1, 0 or 2.

Now between x = -2 and x = 0 and between x = 0 and x = 2 you need to determine "which curve is on top and which is on the bottom", that is which function f or g is larger.

Choose a point between x = -2 and x = 0, say x = -1.

f(-1) = (-1)3 -6(-1) = 5 and
g(-1) = -2(-1) = 2 Thus between -2 and 0 f is larger than g.

Choose a point between x = 0 and x = 2, say x = 1.

f(1) = (1)3 -6(1) = -5 and
g(-1) = -2(1) = -2 Thus between 0 and 2 g is larger than f.

Hence the area between the two curves is Cheers,
Harley
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