53.8k views
18 votes
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

The graph of the function f(x) is shown below. -a If a = 9, then find the value of-example-1
User Rob Rodi
by
7.2k points

1 Answer

10 votes

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]

User Navarasu
by
6.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.