Let `f (x) = x^(2) -2. If int _(3) ^(6) f (x)dx =3f (c )` for some `c in (3,6)` then the value of c is equal to

Options

A. `sqrt12`
B. `sqrt21`
C. `sqrt19`
D. `sqrt17`

Answer

To solve the problem, we need to evaluate the integral of the function \( f(x) = x^2 - 2 \) from 3 to 6 and set it equal to \( 3f(c) \) for some \( c \) in the interval \( (3, 6) \).

### Step-by-Step Solution:

1. **Set up the integral**:
We need to compute the integral:
\[ \int_{3}^{6} (x^2 - 2) \, dx \]

2. **Calculate the integral**:
We can split the integral into two parts:
\[ \int_{3}^{6} x^2 \, dx - \int_{3}^{6} 2 \, dx \]

- For the first part, we use the formula for the integral of \( x^n \):
\[ \int x^2 \, dx = \frac{x^3}{3} \]
Evaluating from 3 to 6:
\[ \left[ \frac{x^3}{3} \right]_{3}^{6} = \frac{6^3}{3} - \frac{3^3}{3} = \frac{216}{3} - \frac{27}{3} = 72 - 9 = 63 \]

- For the second part:
\[ \int 2 \, dx = 2x \quad \text{and evaluating from 3 to 6 gives:} \]
\[ [2x]_{3}^{6} = 2(6) - 2(3) = 12 - 6 = 6 \]

- Combining both results:
\[ \int_{3}^{6} (x^2 - 2) \, dx = 63 - 6 = 57 \]

3. **Set the integral equal to \( 3f(c) \)**:
We have:
\[ 57 = 3f(c) \]
Since \( f(c) = c^2 - 2 \), we can substitute:
\[ 57 = 3(c^2 - 2) \]

4. **Solve for \( c^2 \)**:
Rearranging the equation:
\[ 57 = 3c^2 - 6 \]
\[ 3c^2 = 57 + 6 = 63 \]
\[ c^2 = \frac{63}{3} = 21 \]

5. **Find \( c \)**:
Taking the square root:
\[ c = \sqrt{21} \]

### Final Answer:
Thus, the value of \( c \) is:
\[ c = \sqrt{21} \]