Question

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

-a
If a = 9, then find the value of
3a
f(x) d
-a
f(x) dx =
a
3a
fr f(x) dx.
-a
ROUND YOUR FINAL ANSWER TO 4 DECIMAL PLACES.
2a
3a

Answer 1

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)*81}2 \approx \boxed{5.7345}`[/tex]