The figure to the right shows the areas of regions bounded by the graph of \( f \) and the \( x \)-axis. Evaluate the following integral. \( \int_{\text {a }}^{0} f(x) d x \)

I'm happy to help you evaluate the integral \( \int_{\text{a}}^{0} f(x) \, dx \). However, to provide an accurate solution, I would need more information about the function \( f(x) \) and the specific details presented in the figure you're referring to, such as the areas of the regions bounded by \( f(x) \) and the \( x \)-axis between \( x = \text{a} \) and \( x = 0 \). Here are some general guidelines on how to approach such problems: 1. **Understanding the Integral**: - The integral \( \int_{\text{a}}^{0} f(x) \, dx \) represents the net area between the function \( f(x) \) and the \( x \)-axis from \( x = \text{a} \) to \( x = 0 \). - If \( f(x) \) is above the \( x \)-axis in this interval, the integral gives the positive area. - If \( f(x) \) is below the \( x \)-axis, the integral gives the negative of the area. 2. **Using Given Areas**: - If the figure provides specific areas of regions where \( f(x) \) is positive and/or negative, you can sum these areas accordingly. - For example, if there are two regions where \( f(x) \) is positive with areas \( A_1 \) and \( A_2 \), and one region where \( f(x) \) is negative with area \( A_3 \), then: \[ \int_{\text{a}}^{0} f(x) \, dx = A_1 + A_2 - A_3 \] 3. **Sign Considerations**: - Always pay attention to whether the function is above or below the \( x \)-axis within the limits of integration, as this affects the sign of each area contribution. 4. **Example**: - Suppose the figure shows that between \( x = \text{a} \) and \( x = 0 \), the function \( f(x) \) is above the \( x \)-axis from \( x = \text{a} \) to \( x = \text{b} \) with area \( A \), and below the \( x \)-axis from \( x = \text{b} \) to \( x = 0 \) with area \( B \). Then: \[ \int_{\text{a}}^{0} f(x) \, dx = A - B \] If you can provide more details about the function \( f(x) \) or describe the areas shown in the figure, I'd be able to give a more specific answer. Feel free to share additional information!