Find these on the graph **Transcription for Educational Website**---**Use this figure to find the following definite integrals:****Description of the Graph:**The graph presents a curve labeled $y = f(x)$, and the x-axis and y-axis are shown. Three distinct regions are shaded under the curve, each indicating different definite integrals. - **Yellow Region**: - Located above the x-axis. - Bounded between x-values labeled $0$ and an unlabeled point, presumably positive. - The area of the yellow region is marked as $12$.- **Blue Region**: - Located below the x-axis, indicating a negative integral. - Bounded between x-values labeled $a$ and $b$. - The area of the blue region is marked as $6$.- **Orange Region**: - Located below the x-axis. - Bounded between x-values labeled $b$ and $c$. - The area of the orange region is marked as $11$.These regions can be used to calculate definite integrals over different intervals by considering the areas above and below the x-axis, with areas below indicating negative contributions to the integral. Certainly! Below is a transcription of the text from the image, suitable for an educational website:---### Understanding Definite IntegralsA definite integral is represented in the following forms, where $ f(x) $ is the function being integrated, and $ a $, $ b $, $ c $ are the limits of integration:1. $\int_{0}^{a} f(x) \, dx$2. $\int_{0}^{b} f(x) \, dx$3. $\int_{0}^{c} f(x) \, dx$4. $\int_{b}^{a} f(x) \, dx$5. $\int_{c}^{a} f(x) \, dx$These integrals are used to calculate the area under the curve $ f(x) $ from one limit to another on the x-axis. The notation $\int$ is an integral sign, $ dx $ indicates integration with respect to $ x $.### Detailed Explanation- **$\int_{0}^{a} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = 0 $ to $ x = a $.- **$\int_{0}^{b} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = 0 $ to $ x = b $.- **$\int_{0}^{c} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = 0 $ to $ x = c $.- **$\int_{b}^{a} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = b $ to $ x = a $. It may result in a negative value if $ b > a $.- **$\int_{c}^{a} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = c $ to $ x = a $.These integrals are fundamental concepts in calculus, used extensively in science, engineering, and mathematics to solve various problems involving rates of change and areas.---

Answered: Use this figure to find the following…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Find these on the graph

**Transcription for Educational Website**

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**Use this figure to find the following definite integrals:**

**Description of the Graph:**

The graph presents a curve labeled $y = f(x)$, and the x-axis and y-axis are shown. Three distinct regions are shaded under the curve, each indicating different definite integrals.

- **Yellow Region**:
- Located above the x-axis.
- Bounded between x-values labeled $0$ and an unlabeled point, presumably positive.
- The area of the yellow region is marked as $12$.

- **Blue Region**:
- Located below the x-axis, indicating a negative integral.
- Bounded between x-values labeled $a$ and $b$.
- The area of the blue region is marked as $6$.

- **Orange Region**:
- Located below the x-axis.
- Bounded between x-values labeled $b$ and $c$.
- The area of the orange region is marked as $11$.

These regions can be used to calculate definite integrals over different intervals by considering the areas above and below the x-axis, with areas below indicating negative contributions to the integral.

Certainly! Below is a transcription of the text from the image, suitable for an educational website:

---

### Understanding Definite Integrals

A definite integral is represented in the following forms, where $ f(x) $ is the function being integrated, and $ a $, $ b $, $ c $ are the limits of integration:

1. $\int_{0}^{a} f(x) \, dx$
2. $\int_{0}^{b} f(x) \, dx$
3. $\int_{0}^{c} f(x) \, dx$
4. $\int_{b}^{a} f(x) \, dx$
5. $\int_{c}^{a} f(x) \, dx$

These integrals are used to calculate the area under the curve $ f(x) $ from one limit to another on the x-axis. The notation $\int$ is an integral sign, $ dx $ indicates integration with respect to $ x $.

### Detailed Explanation

- **$\int_{0}^{a} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = 0 $ to $ x = a $.
- **$\int_{0}^{b} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = 0 $ to $ x = b $.
- **$\int_{0}^{c} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = 0 $ to $ x = c $.
- **$\int_{b}^{a} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = b $ to $ x = a $. It may result in a negative value if $ b > a $.
- **$\int_{c}^{a} f(x) \, dx$**: This calculates the area under $ f(x) $ from $ x = c $ to $ x = a $.

These integrals are fundamental concepts in calculus, used extensively in science, engineering, and mathematics to solve various problems involving rates of change and areas.

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Expert Solution

Step 1

The figure given in the problem represents the graph of the function y=f(x).

The region under the curve of the graph represents the values of a function between different integral.

The region about the x-axis is a positive region and the region below the x-axis is a negative region.

For $\int_{0}^{a} f (x) d x$ ,

From the graph, the value of the function in between 0 and a is 12.

So,

$\int_{0}^{a} f (x) d x = 12$

Thus, the value of the definite integral $\int_{0}^{a} f (x) d x$ is 12.

For $\int_{0}^{b} f (x) d x$ ,

From the graph, the value of the function in between 0 and a is 12 and between a and b is 6. As the region between a and b is below x-axis it is the negative region.

So, the integral will be the algebraic sum of values of both regions.

The value of integral is,

$\begin{array}{rcl} \int_{0}^{b} f (x) d x & = & 12 - 6 \\ = & 6 \end{array}$

Thus, the value of the definite integral $\int_{0}^{b} f (x) d x$ is 6.

Step 2

For $\int_{0}^{c} f (x) d x$ ,

From the graph, the value of the function in between 0 and a is 12 (positive), between a and b is 6 (negative), and between b and c is 11 (positive).

So, the integral will be the algebraic sum of values of each region between 0 to c.

The value of integral is,

$\begin{array}{rcl} \int_{0}^{c} f (x) d x & = & 12 - 6 + 11 \\ = & 6 + 11 \\ = & 17 \end{array}$

Thus, the value of the definite integral $\int_{0}^{c} f (x) d x$ is 17.

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