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erica [24]
2 years ago
11

The graph of the function f(x) is shown below.

Mathematics
1 answer:
Zielflug [23.3K]2 years ago
4 0

The integral of f(x) over the interval [-a,a] corresponds to the area of a semicircle of radius a, so

\displaystyle \int_{-a}^a f(x) \, dx = \frac{\pi a^2}2

The integral over [a,3a] corresponds to the negative area of a trapezoid with height a and base lengths 2a and a, so

\displaystyle \int_a^{3a} f(x) \, dx = -\frac{a(2a+a)}2 = -\frac{3a^2}2

Then combining the integrals, we have

\displaystyle \int_{-a}^{3a} f(x) \, dx = \frac{\pi a^2}2 + \left(-\frac{3a^2}2\right) = \frac{(\pi-3)a^2}2

When a=9, the integral evaluates to

\displaystyle \int_{-9}^{27} f(x) \, dx = \frac{(\pi-3)\times81}2 \approx \boxed{5.7345}[/tex]

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