Questions: Find the area bounded by the graphs of the indicated equations over the given interval (when stated). Compute answers to three decimal places. y=x^2+1; y=2x-2; -1 ≤ x ≤ 2 The area, calculated to three decimal places, is square units.

Solution Steps

We are given two functions: \[ f_1(x) = x^2 + 1 \] \[ f_2(x) = 2x - 2 \]

To find the area between the curves, we first need to determine the points where the two functions intersect. We solve the equation: \[ x^2 + 1 = 2x - 2 \] This simplifies to: \[ x^2 - 2x + 3 = 0 \] The solutions to this equation are: \[ x = 1 \pm \sqrt{2}i \] Since the solutions are complex, it indicates that the curves do not intersect in the real number domain over the interval \([-1, 2]\).

Next, we calculate the area between the two curves over the interval \([-1, 2]\). The area \(A\) is given by the integral: \[ A = \int_{-1}^{2} |f_1(x) - f_2(x)| \, dx \] Calculating this integral yields: \[ A = 9 \]

Final Answer