Area under the curve

7.11

Find the total area enclosed by the functions f and g:

\[f(x)=2x^2 +5x−3\]

\[g(x)=x^2 +4x−1\]

\[f(x) - g(x) = (2x^2 +5x−3) - (x^2 +4x−1)\]

\[f(x) - g(x) = 2x^2 +5x − 3 - x^2 - 4x + 1\] \[f(x) - g(x) = x^2 +x − 2\]

To find the area in which to integrate…

\[x^2 +4x−1=2x^2 +5x−3\]

\[0=x^2 +x−2\]

\[(x-1) (x+2)\]

\[ x = 1, x = -2 \]

\[ \int_{-2}^{1} (x^2 +x − 2) dx \]

\[ \int_{-2}^{1} (\dfrac{x^3}{3} + \dfrac{x^2}{2} − 2x) dx \]

\[ \int_{-2}^{1} x(\dfrac{x^2}{3} + \dfrac{x}{2} − 2) dx \]

At x = -2

\[-2 *(\dfrac{4}{3} + \dfrac{-2}{2} -2) = \dfrac{10}{3}\]

At x = 1

\[1(\dfrac{1}{3} + \dfrac{1}{2} − 2)=- \dfrac{7}{6}\]

\[ \dfrac{20}{6} - \dfrac{-7}{6} = 27/6 = 4.5 \]

Area under the curve: 4.5