the function f is shown below. what is the value of ∫-4,3 f(x) dx? write your answer in simplest form.

Answer:

We need to use the property that the definite - integral $\int_{a}^{b}f(x)dx$ is equal to the net - signed area between the curve $y = f(x)$ and the $x$ - axis from $x=a$ to $x = b$. We can split the interval $[-4,3]$ into sub - intervals and find the area of each geometric shape (triangles and trapezoids) formed between the curve and the $x$ - axis.

1. Sub - interval $[-4,-2]$:
   • This part of the graph forms a trapezoid. The formula for the area of a trapezoid is $A=\frac{(b_1 + b_2)h}{2}$, where $b_1$ and $b_2$ are the lengths of the parallel sides and $h$ is the height.
   • For the trapezoid on $[-4,-2]$, $b_1=3$, $b_2 = 4$, and $h = 2$. So the area $A_1=\frac{(3 + 4)\times2}{2}=7$.
2. Sub - interval $[-2,1]$:
   • This part forms a triangle. The formula for the area of a triangle is $A=\frac{1}{2}bh$. Here, $b = 3$ and $h=6$. So the area $A_2=\frac{1}{2}\times3\times6 = 9$.
3. Sub - interval $[1,3]$:
   • This part forms a triangle below the $x$ - axis. The base $b = 2$ and the height $h = 6$. The area of this triangle is $A_3=\frac{1}{2}\times2\times6=6$, but since it is below the $x$ - axis, its contribution to the definite integral is $- 6$.
4. Calculate the definite integral:
   • $\int_{-4}^{3}f(x)dx=A_1 + A_2+A_3$.
   • $\int_{-4}^{3}f(x)dx=7 + 9-6$.
   • $\int_{-4}^{3}f(x)dx = 10$.

Explanation:

Step1: Identify geometric shapes

Identify trapezoid and triangles on $[-4,3]$.

Step2: Calculate area of trapezoid on $[-4,-2]$

Use $A=\frac{(b_1 + b_2)h}{2}$, $A_1=\frac{(3 + 4)\times2}{2}=7$.

Step3: Calculate area of triangle on $[-2,1]$

Use $A=\frac{1}{2}bh$, $A_2=\frac{1}{2}\times3\times6 = 9$.

Step4: Calculate area of triangle on $[1,3]$

Use $A=\frac{1}{2}bh$, $A_3=\frac{1}{2}\times2\times6 = 6$ (negative as below $x$ - axis).

Step5: Sum up the areas

$\int_{-4}^{3}f(x)dx=7 + 9-6=10$.