https://file.aiquickdraw.com/mathbot/file/1721187545956340mz771.png

To find the area between the curves given by \( y = e^x \) and \( y = 5 - e^x \) between the vertical lines \( x = -4 \) and \( x = 3 \), we need to set up the integral(s).

First, we recognize that the area between two curves \( y = f(x) \) and \( y = g(x) \) from \( x = a \) to \( x = b \) is given by the integral:

\[ \text{Area} = \int_a^b |f(x) - g(x)| \, dx \]

In this case, \( f(x) = 5 - e^x \) and \( g(x) = e^x \). Since \( 5 - e^x \) is always greater than \( e^x \) in the interval \( x \in [-4, 3] \), we can write:

\[ \text{Area} = \int_{-4}^{3} [(5 - e^x) - e^x] \, dx \]
\[ \text{Area} = \int_{-4}^{3} (5 - 2e^x) \, dx \]

Now, let's evaluate the options given in the image.

### Option A
\[ \int_{-4}^{\boxed{}} 5 - 2e^x \, dx + \int_{\boxed{}}^3 5 - 2e^x \, dx \]

This is incorrect because it suggests splitting the integral unnecessarily.

### Option B
\[ \int_{-4}^{3} 5 - 2e^x \, dx \]

This is the correct representation. 

Thus, the correct choice is **B**.

### Solution:

To solve the integral:

\[ \int_{-4}^{3} (5 - 2e^x) \, dx \]

1. Separate the integral:
\[ \int_{-4}^{3} 5 \, dx - 2 \int_{-4}^{3} e^x \, dx \]

2. Evaluate each integral separately:
\[ \int_{-4}^{3} 5 \, dx = 5[x]_{-4}^{3} = 5(3 - (-4)) = 5(7) = 35 \]

\[ \int_{-4}^{3} e^x \, dx = e^x \bigg|_{-4}^{3} = e^3 - e^{-4} \]

3. Combine the results:
\[ \text{Area} = 35 - 2(e^3 - e^{-4}) = 35 - 2e^3 + 2e^{-4} \]

Therefore, the area between the curves is:

\[ \boxed{35 - 2e^3 + 2e^{-4}} \]

---

Would you like more details or have any questions about this solution? Here are some related questions you might want to ask:

1. How do we determine which curve is above the other?
2. Can you explain how to set up an integral for the area between curves?
3. How do we handle integrals with more complex bounds or curves?
4. What if the curves intersect within the interval?
5. Can you show a step-by-step integration process for similar problems?

**Tip:** When finding the area between curves, always sketch the graph first to determine which function is on top and to correctly set up the limits of integration.

Discover how to calculate the area between the curves y = e^x and y = 5 - e^x within the interval x = -4 to x = 3. This detailed guide covers the integral setup and evaluation, resulting in the area expression 35 - 2e^3 + 2e^{-4}. Ideal for high school calculus 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