Answer:
35.5
Step-by-step explanation:
The integral ∫(-9 to 9) f(x) dx represents the area under the curve of the function f from x=-9 to x=9. Given the piecewise linear function defined by the points you provided, this area can be calculated as the sum of the areas of the trapezoids and triangles formed by these points.
Let’s calculate the areas of these shapes:
Trapezoid between (-9,4), (-6,5): Area = 1/2 * |x₂ - x₁| * (y₁ + y₂) = 1/2 * 3 * (4 + 5) = 13.5
Triangle between (-6,5), (-3,0): Area = 1/2 * base * height = 1/2 * 3 * 5 = 7.5
Trapezoid between (-3,0), (0,2): Area = 1/2 * 3 * (0 + 2) = 3
Triangle between (0,2), (4,0): Area = 1/2 * 4 * 2 = 4
Trapezoid between (4,0), (6,3): Area = 1/2 * 2 * (0 + 3) = 3
Triangle between (6,3), (9,0): Area = 1/2 * 3 * 3 = 4.5
Adding these areas together gives the total area under the curve, which is the value of the integral:
Total Area = 13.5 + 7.5 + 3 + 4 + 3 + 4.5 = 35.5
So, the value of ∫(-9 to 9) f(x) dx is 35.5. This is your answer in simplest form.
