Question

In a rough sketch, mark the region bounded by \( y = 1 + |x + 1| \), \( x = -2 \), \( x = 2 \), and \( y = 0 \). Using integration, find the area of the marked region.

Hint:

To find the area under a curve with an absolute value function, split the integral at the point where the expression inside the absolute value changes sign.

Answer 1

The objective is to determine the area bounded by the curve \( y = 1 + |x+1| \) and the x-axis, for \( x \) values ranging from \( -2 \) to \( 2 \). The absolute value function necessitates splitting the calculation into two intervals. For the interval \( [-2, -1] \), the expression simplifies to \( y = -x \). For the interval \( [-1, 2] \), the expression simplifies to \( y = x + 2 \). The area is computed by integrating these simplified functions over their respective intervals. The area for the first interval \( [-2, -1] \) is \( A_1 = \int_{-2}^{-1} (-x) dx = \frac{3}{2} \). The area for the second interval \( [-1, 2] \) is \( A_2 = \int_{-1}^{2} (x+2) dx = \frac{15}{2} \). The total area is the sum of these two areas: \( A = A_1 + A_2 = \frac{3}{2} + \frac{15}{2} = 9 \). \begin{center} \includegraphics[width=0.7\textwidth]{ig4.png} \end{center}