The Function $f$ Is Shown Below. What Is The Value Of $\int_{-7}^4 F(x) \, Dx$? Write Your Answer In Simplest Form.

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Introduction

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In this article, we will explore the function $f$ and calculate the value of its definite integral from $-7$ to $4$. The function $f$ is given, and we will use various mathematical techniques to find the integral value.

The Function $f$

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The function $f$ is defined as:

$$f(x) = \begin{cases} 3x + 2 & \text{if } x < -2 \\ 5 - 2x & \text{if } x \geq -2 \end{cases}$$

Calculating the Integral

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To calculate the integral of $f(x)$ from $-7$ to $4$, we need to break it down into two parts: the integral from $-7$ to $-2$ and the integral from $-2$ to $4$.

Integral from $-7$ to $-2$

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For the integral from $-7$ to $-2$, we use the function $f(x) = 3x + 2$.

$$\int_{-7}^{-2} f(x) \, dx = \int_{-7}^{-2} (3x + 2) \, dx$$

Using the power rule of integration, we get:

$$\int_{-7}^{-2} (3x + 2) \, dx = \left[ \frac{3x^2}{2} + 2x \right]_{-7}^{-2}$$

Evaluating the expression at the limits, we get:

$$\left[ \frac{3x^2}{2} + 2x \right]_{-7}^{-2} = \left( \frac{3(-2)^2}{2} + 2(-2) \right) - \left( \frac{3(-7)^2}{2} + 2(-7) \right)$$

Simplifying the expression, we get:

$$\left( \frac{3(-2)^2}{2} + 2(-2) \right) - \left( \frac{3(-7)^2}{2} + 2(-7) \right) = \left( \frac{12}{2} - 4 \right) - \left( \frac{147}{2} - 14 \right)$$

$$= (6 - 4) - \left( \frac{147}{2} - 14 \right)$$

$$= 2 - \left( \frac{147}{2} - 14 \right)$$

$$= 2 - \frac{147}{2} + 14$$

$$= 2 - \frac{147}{2} + \frac{28}{2}$$

$$= 2 - \frac{119}{2}$$

$$= \frac{4}{2} - \frac{119}{2}$$

$$= -\frac{115}{2}$$

Integral from $-2$ to $4$

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For the integral from $-2$ to $4$, we use the function $f(x) = 5 - 2x$.

$$\int_{-2}^{4} f(x) \, dx = \int_{-2}^{4} (5 - 2x) \, dx$$

Using the power rule of integration, we get:

$$\int_{-2}^{4} (5 - 2x) \, dx = \left[ 5x - x^2 \right]_{-2}^{4}$$

Evaluating the expression at the limits, we get:

$$\left[ 5x - x^2 \right]_{-2}^{4} = \left( 5(4) - (4)^2 \right) - \left( 5(-2) - (-2)^2 \right)$$

Simplifying the expression, we get:

$$\left( 5(4) - (4)^2 \right) - \left( 5(-2) - (-2)^2 \right) = \left( 20 - 16 \right) - \left( -10 - 4 \right)$$

$$= (4) - (-14)$$

$$= 4 + 14$$

$$= 18$$

Combining the Integrals

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Now that we have calculated the two integrals separately, we can combine them to get the final answer.

$$\int_{-7}^{4} f(x) \, dx = \int_{-7}^{-2} f(x) \, dx + \int_{-2}^{4} f(x) \, dx$$

$$= -\frac{115}{2} + 18$$

$$= -\frac{115}{2} + \frac{36}{2}$$

$$= -\frac{79}{2}$$

The final answer is $\boxed{-\frac{79}{2}}$.

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Introduction

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In our previous article, we explored the function $f$ and calculated the value of its definite integral from $-7$ to $4$. In this article, we will answer some frequently asked questions related to the function $f$ and its integral value.

Q&A

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Q: What is the function $f$?

A: The function $f$ is defined as:

$$f(x) = \begin{cases} 3x + 2 & \text{if } x < -2 \\ 5 - 2x & \text{if } x \geq -2 \end{cases}$$

Q: How do I calculate the integral of $f(x)$ from $-7$ to $4$?

A: To calculate the integral of $f(x)$ from $-7$ to $4$, you need to break it down into two parts: the integral from $-7$ to $-2$ and the integral from $-2$ to $4$. For the integral from $-7$ to $-2$, you use the function $f(x) = 3x + 2$. For the integral from $-2$ to $4$, you use the function $f(x) = 5 - 2x$.

Q: What is the value of the integral from $-7$ to $-2$?

A: The value of the integral from $-7$ to $-2$ is:

$$\int_{-7}^{-2} f(x) \, dx = -\frac{115}{2}$$

Q: What is the value of the integral from $-2$ to $4$?

A: The value of the integral from $-2$ to $4$ is:

$$\int_{-2}^{4} f(x) \, dx = 18$$

Q: What is the final answer to the integral from $-7$ to $4$?

A: The final answer to the integral from $-7$ to $4$ is:

$$\int_{-7}^{4} f(x) \, dx = -\frac{79}{2}$$

Q: Can I use a calculator to calculate the integral?

A: Yes, you can use a calculator to calculate the integral. However, it's always a good idea to understand the underlying math and be able to calculate the integral by hand.

Q: What if I have a different function $f$?

A: If you have a different function $f$, you will need to follow the same steps to calculate the integral. However, the specific steps and calculations will depend on the function $f$.

Conclusion

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In this article, we answered some frequently asked questions related to the function $f$ and its integral value. We hope that this article has been helpful in understanding the function $f$ and its integral value.

Additional Resources

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• Introduction to Calculus
• Definite Integrals
• Power Rule of Integration

We hope that this article has been helpful in understanding the function $f$ and its integral value. If you have any further questions, please don't hesitate to ask.