Chapter 7, Pg 360 Question 13 Find the total area enclosed by the
functions f and g.
\(f(x) = 2x^2 + 5x -3\), \(g(x) = x^2 + 4x -1\)
To find the area of the region bounded by the two functions, we first need to find the points where the two equations meet by setting the equations equal to one another.
\[ 2x^{2} + 5x − 3 = x^2 + 4x -1 \\ x^{2} + x − 2 = 0 \\ (x + 2)(x - 1) = 0 \\ x = 1, -2 \] Therefore upper limit of the integral is 1 while the lower limit is -2
\[ \int_{-2}^{1} (f(x) - g(x))dx \\ = \int_{-2}^{1} [(2x^{2} + 5x -3) - (x^2 + 4x -1)]dx \\ = \int_{-1}^{4}(x^{2} + x - 2)dx \\ = [\frac{x^{3}}{3} + \frac{x^{2}}{2} - 2x]_{-2}^{1} \\ = (\frac{1}{3} + \frac{1}{2} - 2) - (\frac{-8}{3} + \frac{4}{2} + 4)\\ = -4.5 \] Since area cannot be negative, the total area enclosed by functions f and g is 4.5 sq units.