Answer: 4.5
Step-by-step explanation:
First, find the points of intersection by solving the system.
y = x² + 2x + 4
y = x + 6
Solve by substitution:
x² + 2x + 4 = x + 6 ⇒ x² + x - 2 = 0 ⇒ (x + 2)(x - 1) = 0 ⇒ x = -2, x = 1
Now, integrate from x = -2 to x = 1
[tex]\int\limits^1_2 {(x+6)-(x^{2}+2x+4) } \,[/tex] the bottom of the integral is -2
= [tex]\int\limits^1_2 {x+6-x^{2}-2x-4 } \,[/tex]
= [tex]\int\limits^1_2 {-x^{2}-x+2 } \,[/tex]
= [tex]\frac{-x^{3}}{3} - \frac{x^{2}}{2}+2x\int\limits^1_2 {} \,[/tex]
= [tex](\frac{-1^{3}}{3} - \frac{1^{2}}{2}+2(1)) - (\frac{-(-2)^{3}}{3} - \frac{(-2)^{2}}{2}+2(-2))[/tex]
= [tex](\frac{-1}{3} - \frac{1}{2} +2) - (\frac{8}{3} -\frac{4}{2} -4)[/tex]
= [tex]\frac{-9}{3} + \frac{3}{2} +6[/tex]
= -3 + 1.5 + 6
= 4.5
