Questions: The function f is shown below. What is the value of ∫ from -7 to 1 of f(x) dx? Write your answer in simplest form.

Solution Steps

The area under the curve from $x=-7$ to $x=1$ can be divided into three geometric shapes:

1. A triangle from $x=-9$ to $x=-7$ with base 2 and height 2.
2. A triangle from $x=-7$ to $x=-3$ with base 4 and height 8.
3. A trapezoid from $x=-3$ to $x=1$ with bases 6 and 2, and height 4.

1. The area of the first triangle is $\frac{1}{2}(2)(2) = 2$.
2. The area of the second triangle is $\frac{1}{2}(4)(8) = 16$.
3. The area of the trapezoid from $x=-3$ to $x=0$ is $\frac{1}{2}(6+2)(3) = \frac{1}{2}(8)(3) = 12$. The area of the rectangle from $x=0$ to $x=1$ is $(1)(-2) = -2$. The trapezoid from $x=-3$ to $x=1$ is above the $x$-axis from $x=-3$ to $x=0$, and it is below the $x$-axis from $x=0$ to $x=1$. So its area is the difference of the areas of the triangle above the x-axis minus the rectangle below the x-axis. The area of the triangle above the x-axis is $1/2 \cdot (3) \cdot 6 = 9$. The area of the rectangle below the x-axis is $|-2| = 2$. So, the area of the trapezoid is $9 - 2 = 10$ Another way to calculate the trapezoid is $1/2 (6 + (-2)) \cdot 4 = 1/2 \cdot 4 \cdot 4 = 8$. The total area under the curve is $2 + 16 + 8 = 26$.

The integral of the function $f(x)$ from $x=-7$ to $x=1$ is the sum of the areas of the shapes calculated above. Since the areas are signed areas, we simply add them:

$\int_{-7}^1 f(x) dx = 2 + 16 + 8 = 26$

Final Answer