Find the area of the region bounded by the graph of \( f \) and the \( x \)-axis on the given interval. \( f(x)=\frac{4}{x} \) on \( [-2,-1] \) The area of the region is (Type an exact answer.)

To find the area of the region bounded by the graph of \( f(x) = \frac{4}{x} \) and the \( x \)-axis on the interval \( [-2, -1] \), we need to calculate the definite integral of \( f(x) \) from \( -2 \) to \( -1 \). The area \( A \) can be calculated using the following integral: \[ A = \int_{-2}^{-1} f(x) \, dx = \int_{-2}^{-1} \frac{4}{x} \, dx \] Now, let's compute the integral step by step. 1. **Find the antiderivative of \( f(x) \)**: The antiderivative of \( \frac{4}{x} \) is \( 4 \ln |x| \). 2. **Evaluate the definite integral**: We will evaluate \( 4 \ln |x| \) from \( -2 \) to \( -1 \): \[ A = \left[ 4 \ln |x| \right]_{-2}^{-1} \] 3. **Calculate the values**: - At \( x = -1 \): \[ 4 \ln |-1| = 4 \ln 1 = 0 \] - At \( x = -2 \): \[ 4 \ln |-2| = 4 \ln 2 \] 4. **Combine the results**: Now, substituting these values into the definite integral: \[ A = 0 - 4 \ln 2 = -4 \ln 2 \] Since area cannot be negative, we take the absolute value: \[ A = 4 \ln 2 \] Thus, the area of the region bounded by the graph of \( f \) and the \( x \)-axis on the interval \( [-2, -1] \) is \[ \boxed{4 \ln 2} \]