IF `int_(-3)^2 f(x) dx=7/3 and int_(-3)^9 f(x) dx=-5/6` then what is the value of ` int_(2)^(9)f(x) dx`

Options

A. `-19/6`
B. `19/6`
C. `3/2`
D. `-3/2`

Answer

To solve the problem, we need to find the value of the integral \( \int_{2}^{9} f(x) \, dx \) given the following information:

1. \( \int_{-3}^{2} f(x) \, dx = \frac{7}{3} \)
2. \( \int_{-3}^{9} f(x) \, dx = -\frac{5}{6} \)

### Step-by-step Solution:

1. **Understand the relationship between the integrals**:
We know that:
\[ \int_{-3}^{9} f(x) \, dx = \int_{-3}^{2} f(x) \, dx + \int_{2}^{9} f(x) \, dx \]
This is based on the property of definite integrals, which states that the integral over an interval can be split into the sum of integrals over subintervals.

2. **Substituting the known values**:
From the problem, we can substitute the known values into the equation:
\[ -\frac{5}{6} = \frac{7}{3} + \int_{2}^{9} f(x) \, dx \]

3. **Isolate the integral**:
To find \( \int_{2}^{9} f(x) \, dx \), we need to isolate it on one side of the equation:
\[ \int_{2}^{9} f(x) \, dx = -\frac{5}{6} - \frac{7}{3} \]

4. **Finding a common denominator**:
The denominators are 6 and 3. The least common multiple is 6. We can rewrite \( \frac{7}{3} \) with a denominator of 6:
\[ \frac{7}{3} = \frac{14}{6} \]

5. **Perform the subtraction**:
Now we can perform the subtraction:
\[ \int_{2}^{9} f(x) \, dx = -\frac{5}{6} - \frac{14}{6} = -\frac{5 + 14}{6} = -\frac{19}{6} \]

6. **Final answer**:
Thus, the value of \( \int_{2}^{9} f(x) \, dx \) is:
\[ \int_{2}^{9} f(x) \, dx = -\frac{19}{6} \]

### Summary:
The value of \( \int_{2}^{9} f(x) \, dx \) is \( -\frac{19}{6} \).