Questions: Find the area of the region that is bounded above by the curve f(x)=(x+2)^2 and the line g(x)=-x and bounded below by the x-axis. It can be helpful to graph the functions to better determine the correct region. Enter an exact answer.

Solution Steps

We define the functions that bound the region of interest:

• The curve is given by \( f(x) = (x + 2)^2 \).
• The line is given by \( g(x) = -x \).

To find the points where the curve and the line intersect, we solve the equation: \[ f(x) = g(x) \implies (x + 2)^2 = -x \] This results in the intersection points \( x = -4 \) and \( x = -1 \).

The area \( A \) between the curve and the line from \( x = -4 \) to \( x = -1 \) is given by the integral: \[ A = \int_{-4}^{-1} \left( f(x) - g(x) \right) \, dx = \int_{-4}^{-1} \left( (x + 2)^2 - (-x) \right) \, dx \]

Evaluating the integral, we find: \[ A = \int_{-4}^{-1} \left( (x + 2)^2 + x \right) \, dx \] After performing the integration, we find that the area is \( A = -\frac{9}{2} \). Since area cannot be negative, we take the absolute value, resulting in the final area of the bounded region.

Final Answer