Sketch the graph of \( y = |x + 3| \) and find the area of the region enclosed by the curve, x-axis, between \( x = -6 \) and \( x = 0 \), using integration.
Show Hint
When finding the area between a curve and the x-axis, break the integral into parts if the function has different expressions for different intervals, especially for absolute value functions.
The given function is \( y = |x + 3| \). The absolute value function splits into two cases:
\[
y = \begin{cases}
x + 3 & \text{if} \, x \geq -3,\\
-(x + 3) & \text{if} \, x<-3.
\end{cases}
\]
Now, we are asked to find the area between \( x = -6 \) and \( x = 0 \). To do this, we split the integral at \( x = -3 \), since the function has different expressions in these two regions.
1. For \( x \in [-6, -3] \), the equation becomes \( y = -(x + 3) \). The integral in this range is:
\[
A_1 = \int_{-6}^{-3} -(x + 3) \, dx.
\]
2. For \( x \in [-3, 0] \), the equation becomes \( y = x + 3 \). The integral in this range is:
\[
A_2 = \int_{-3}^{0} (x + 3) \, dx.
\]
Now, we will calculate these integrals:
For \( A_1 \):
\[
A_1 = \int_{-6}^{-3} -(x + 3) \, dx = - \left[ \frac{x^2}{2} + 3x \right]_{-6}^{-3}
\]
\[
= - \left[ \left( \frac{(-3)^2}{2} + 3(-3) \right) - \left( \frac{(-6)^2}{2} + 3(-6) \right) \right]
\]
\[
= - \left[ \left( \frac{9}{2} - 9 \right) - \left( \frac{36}{2} - 1 \right) \right]
\]
\[
= - \left[ \left( \frac{9}{2} - \frac{1}{2} \right) - \left( \frac{36}{2} - \frac{36}{2} \right) \right]
\]
\[
= - \left[ -\frac{9}{2} - 0 \right] = \frac{9}{2}.
\]
For \( A_2 \):
\[
A_2 = \int_{-3}^{0} (x + 3) \, dx = \left[ \frac{x^2}{2} + 3x \right]_{-3}^{0}
\]
\[
= \left[ \left( \frac{(0)^2}{2} + 3(0) \right) - \left( \frac{(-3)^2}{2} + 3(-3) \right) \right]
\]
\[
= \left[ 0 - \left( \frac{9}{2} - 9 \right) \right]
\]
\[
= \left[ 0 - \left( \frac{9}{2} - \frac{1}{2} \right) \right]
\]
\[
= \left[ 0 + \frac{9}{2} \right] = \frac{9}{2}.
\]
Thus, the total area is:
\[
A = A_1 + A_2 = \frac{9}{2} + \frac{9}{2} = 9.
\]
Thus, the area of the region enclosed by the curve and the x-axis between \( x = -6 \) and \( x = 0 \) is 9.
