question find the net signed area between the...

# Explanation: ## Step1: Recall the formula for net - signed area The net - signed area $A$ between the graph of $y = f(x)$ and the $x$ - axis over the interval $[a,b]$ is given by $A=\int_{a}^{b}f(x)dx$. Here, $a=-6$, $b = 4$ and $f(x)=x + 2$. So, $A=\int_{-6}^{4}(x + 2)dx$. ## Step2: Use the integral sum rule By the sum rule of integration $\int_{a}^{b}(u(x)+v(x))dx=\int_{a}^{b}u(x)dx+\int_{a}^{b}v(x)dx$. So, $\int_{-6}^{4}(x + 2)dx=\int_{-6}^{4}x dx+\int_{-6}^{4}2dx$. ## Step3: Calculate $\int_{-6}^{4}x dx$ The antiderivative of $x$ is $\frac{1}{2}x^{2}$. Using the fundamental theorem of calculus $\int_{-6}^{4}x dx=\left[\frac{1}{2}x^{2}\right]_{-6}^{4}=\frac{1}{2}(4)^{2}-\frac{1}{2}(-6)^{2}=\frac{16}{2}-\frac{36}{2}=8 - 18=-10$. ## Step4: Calculate $\int_{-6}^{4}2dx$ The antiderivative of the constant function $2$ is $2x$. Using the fundamental theorem of calculus $\int_{-6}^{4}2dx=[2x]_{-6}^{4}=2\times4-2\times(-6)=8 + 12 = 20$. ## Step5: Find the net - signed area $A=\int_{-6}^{4}x dx+\int_{-6}^{4}2dx=-10 + 20=10$. # Answer: $10$