Answer:
The bounded area is 5 + 5/6 square units. (or 35/6 square units)
Step-by-step explanation:
Suppose we want to find the area bounded by two functions f(x) and g(x) in a given interval (x1, x2)
Such that f(x) > g(x) in the given interval.
This area then can be calculated as the integral between x1 and x2 for f(x) - g(x).
We want to find the area bounded by:
f(x) = y = x^2 + 1
g(x) = y = x
x = -1
x = 2
To find this area, we need to f(x) - g(x) between x = -1 and x = 2
This is:
[tex]\int\limits^2_{-1} {(f(x) - g(x))} \, dx[/tex]
[tex]\int\limits^2_{-1} {(x^2 + 1 - x)} \, dx[/tex]
We know that:
[tex]\int\limits^{}_{} {x} \, dx = \frac{x^2}{2}[/tex]
[tex]\int\limits^{}_{} {1} \, dx = x[/tex]
[tex]\int\limits^{}_{} {x^2} \, dx = \frac{x^3}{3}[/tex]
Then our integral is:
[tex]\int\limits^2_{-1} {(x^2 + 1 - x)} \, dx = (\frac{2^3}{2} + 2 - \frac{2^2}{2}) - (\frac{(-1)^3}{3} + (-1) - \frac{(-1)^2}{2} )[/tex]
The right side is equal to:
[tex](4 + 2 - 2) - ( -1/3 - 1 - 1/2) = 4 + 1/3 + 1 + 1/2 = 5 + 2/6 + 3/6 = 5 + 5/6[/tex]
The bounded area is 5 + 5/6 square units.
