Question

evaluate the integral below by interpreting it in terms of areas in the figure. the areas of the labeled regions are a1 = 5, a2 = 3, a3 = 1 and a4 = 2. $v = \int_{5}^{10} f(x) dx$

Explanation:

Step1: Identify regions in integral bounds

The integral $\int_{5}^{10} f(x) dx$ corresponds to regions A3 (above x-axis) and A4 (below x-axis) on the interval $[5,10]$.

Step2: Assign sign based on position

Areas above x-axis are positive, below are negative:
$\int_{5}^{10} f(x) dx = A3 - A4$

Step3: Substitute given area values

Substitute $A3=1$, $A4=2$:
$\int_{5}^{10} f(x) dx = 1 - 2$

Step4: Calculate final value

Compute the subtraction:
$1 - 2 = -1$

Answer:

$-1$