Question
The figure to the right shows the areas of regions bounded by the graph of \( f \) and the \( x \)-axis. Evaluate the following integral. \( \int_{\text {a }}^{0} f(x) d x \)
Ask by Valdez Guerrero. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
To evaluate the integral \( \int_{\text{a}}^{0} f(x) \, dx \), you need to know the specific areas of the regions where \( f(x) \) is above or below the \( x \)-axis between \( x = \text{a} \) and \( x = 0 \). Sum the positive areas and subtract the negative areas to find the net area.
For example, if the function 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:
\[
\int_{\text{a}}^{0} f(x) \, dx = A - B
\]
Without specific information about the function or the areas in the figure, I can't provide a numerical answer.
Solution
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!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To evaluate the integral \( \int_{a}^{0} f(x) \, dx \), you should first consider the properties of definite integrals. If \( f(x) \) has areas below the x-axis, the integral will give a negative value. This means that if you can determine the area from \( a \) to 0, you may need to take the signs into account depending on whether the curve is above or below the x-axis. Imagine breaking down the integral into pieces based on where \( f \) intersects with the x-axis. If \( f \) has regions which are positive and negative over the interval, you can calculate separately, combine the areas while applying appropriate signs, and this will aid in finding the correct value of the integral.
