Refer to BC CALC Volumes of Revolutions:

Question

Refer to BC CALC Volumes of Revolutions: Given f(x)=x^2-3 g(x)=4x+9,

at what values of x does f(x)=g(x)?

Also, find the area enclosed by f(x) and g(x).

Make sure to show all work.

Answer 1

First to calculate all x satisfying

f(x) = g(x)

Let

h(x) = f(x) - g(x) = (x² - 3) - (4x + 9) = x² - 4x - 12

Then we set h(x) = 0, which gives the quadratic equation

x² - 4x - 12 = 0

There are two solutions to this quadratic equation:

x = -2, x = 6

The area enclosed by f(x) and g(x) is the absolute value of

the definite integral between -2 and 6

∫h(x)dx

Let H(x) = ∫h(x)dx =

∫(x² - 4x - 12)dx =

x³/3 - 2x² - 12x + C

Then the area enclosed by f(x) and g(x) is

|H(6) - H(-2)| = |(6³/3 - 2×6² - 12×6) - ((-2)³/3 - 2×(-2)² - 12×(-2))| = 85.33