https://file.aiquickdraw.com/mathbot/file/17327741697686pdwzswd.jpg

The problem states:

If \(\int_{-1}^2 g(x) \, dx = 4\) and \(\int_{-1}^2 3g(x) \, dx = 6\), evaluate \(\int_2^{-1} g(x) \, dx\).

### Solution:

1. **Property of Constants in Integration**:
   Given \(\int_{-1}^2 3g(x) \, dx = 6\), we can factor out the constant 3:
   \[
   3 \int_{-1}^2 g(x) \, dx = 6
   \]
   Divide by 3:
   \[
   \int_{-1}^2 g(x) \, dx = 2
   \]

2. **Relation Between Limits of Integration**:
   By definition of definite integrals, reversing the limits changes the sign:
   \[
   \int_2^{-1} g(x) \, dx = -\int_{-1}^2 g(x) \, dx
   \]

3. **Substitute the Value**:
   From Step 1, \(\int_{-1}^2 g(x) \, dx = 2\). Therefore:
   \[
   \int_2^{-1} g(x) \, dx = -2
   \]

### Final Answer:
\[
\boxed{-2}
\]

Let me know if you have further questions or would like details!  

Here are some related questions to expand understanding:
1. What is the interpretation of a definite integral's sign in terms of the area under the curve?
2. How does multiplying the integrand by a constant affect the definite integral?
3. Can you demonstrate reversing the limits for any definite integral example?
4. How does symmetry in the function \(g(x)\) affect the evaluation of definite integrals?
5. What happens if the bounds of integration are the same?

**Tip**: Always double-check your limits of integration when working with definite integrals, as reversing them changes the sign!

If \(\int_{-1}^2 g(x) \, dx = 4\) and \(\int_{-1}^2 3g(x) \, dx = 6\), evaluate \(\int_2^{-1} g(x) \, dx\).

Learn how to evaluate \(\int_2^{-1} g(x) \, dx\) using properties of definite integrals, such as reversing limits and handling constants. Explore concepts of definite integration and their applications. Suitable for advanced high school or early university students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation