To solve the problem, we need to analyze the given integrals and find the relationships between the variables \( a \) and \( b \).
### Step 1: Analyze the first integral
We are given:
\[
\int_a^b |\sin x| \, dx = 8
\]
The integral of \( |\sin x| \) over one complete period \( [0, \pi] \) is:
\[
\int_0^\pi |\sin x| \, dx = 2
\]
This means that over the interval \( [0, \pi] \), the area under \( |\sin x| \) is 2. To find how many complete periods fit into the interval \( [a, b] \), we can express the integral as:
\[
\int_a^b |\sin x| \, dx = n \cdot 2
\]
where \( n \) is the number of complete periods. Setting this equal to 8, we have:
\[
n \cdot 2 = 8 \implies n = 4
\]
Thus, the interval \( [a, b] \) spans 4 complete periods of \( |\sin x| \), which corresponds to:
\[
b - a = 4\pi
\]
This gives us our first equation:
\[
b - a = 4\pi \quad \text{(Equation 1)}
\]
### Step 2: Analyze the second integral
Next, we have:
\[
\int_0^{a+b} |\cos x| \, dx = 9
\]
The integral of \( |\cos x| \) over one complete period \( [0, \pi] \) is also:
\[
\int_0^\pi |\cos x| \, dx = 2
\]
Thus, we can express the integral as:
\[
\int_0^{a+b} |\cos x| \, dx = m \cdot 2
\]
where \( m \) is the number of complete periods. Setting this equal to 9, we have:
\[
m \cdot 2 = 9 \implies m = \frac{9}{2} = 4.5
\]
This means that the interval \( [0, a+b] \) spans 4.5 periods of \( |\cos x| \), which corresponds to:
\[
a + b = 4.5\pi
\]
This gives us our second equation:
\[
a + b = 4.5\pi \quad \text{(Equation 2)}
\]
### Step 3: Solve the system of equations
Now, we have two equations:
1. \( b - a = 4\pi \)
2. \( a + b = 4.5\pi \)
We can solve this system of equations.
From Equation 1, we can express \( b \) in terms of \( a \):
\[
b = a + 4\pi
\]
Substituting this into Equation 2:
\[
a + (a + 4\pi) = 4.5\pi
\]
This simplifies to:
\[
2a + 4\pi = 4.5\pi
\]
Subtracting \( 4\pi \) from both sides:
\[
2a = 0.5\pi \implies a = \frac{0.5\pi}{2} = \frac{\pi}{4}
\]
Now substituting \( a \) back into the expression for \( b \):
\[
b = \frac{\pi}{4} + 4\pi = \frac{\pi}{4} + \frac{16\pi}{4} = \frac{17\pi}{4}
\]
### Final Values
Thus, we find:
\[
a = \frac{\pi}{4}, \quad b = \frac{17\pi}{4}
\]
