#67 ---### Calculus: Areas of Regions**66–68. Areas of Regions**Find the area of the following regions.\[ x - x^2 \]---**Explanation:**In this exercise, you'll need to determine the area of the regions described by the given mathematical functions. The function you see, $ x - x^2 $, represents a parabola. Typically, for such problems, you will find the area under the curve between specified limits using integration.To find the area of the region bounded by the curve $ y = x - x^2 $ and the x-axis, you can:1. **Find the roots of the equation** $ x - x^2 = 0 $: - Solve for $ x $: $ x(x - 1) = 0 $ So, $ x = 0 $ or $ x = 1 $.2. **Set up the definite integral**: - The integral will be from 0 to 1 (the roots of the equation): \[ \int_{0}^{1} (x - x^2) \, dx \]3. **Evaluate the integral**: - Perform the integration and simplify to find the area.### Next StepsTo proceed, you should:1. Solve the integral using standard integration techniques.2. Verify your results by checking the points of intersection and ensuring the integral boundaries are correct.These problems are fundamental in understanding how to calculate areas using integrals and are very common in calculus courses.--- ### Integral Calculus Problems for Area Determination**Problem 66:** Determine the area of the region bounded by the curve $ y = \frac{x - x^2}{(x + 1)(x^2 + 1)} $ and the x-axis from $ x = 0 $ to $ x = 1 $.**Problem 67:** Determine the area of the region bounded by the curve $ y = \frac{10}{x^2 - 2x - 24} $, the x-axis, and the vertical lines $ x = -2 $ and $ x = 2 $.**Problem 68:** Determine the area of the region in the first quadrant bounded by the curves $ y = \frac{3x^2 + 2x + 1}{x(x^2 + x + 1)} $, $ y = \frac{2}{x} $, and the vertical line $ x = 2 $.### Explanation of Mathematical Concepts and MethodsThese problems involve finding the area of the regions bounded by given curves and specific lines or axes. To find these areas, one needs to set up definite integrals based on the given boundaries and the functions provided.#### Steps to Solve:1. **Identify the boundaries** of the area you need to find. This could involve particular functions, the x-axis, the y-axis, and vertical/horizontal lines.2. **Set up the definite integral(s)** for the functions within the specified boundaries. This involves determining the points of intersection if necessary.3. **Integrate the function(s)** within the limits to find the area.### Detailed Explanation of the Graphs:**Graphs Representation:**1. **Problem 66:** - The curve $ y = \frac{x - x^2}{(x + 1)(x^2 + 1)} $ bounds the region on one side. - The x-axis bounds the region on another side from $ x = 0 $ to $ x = 1 $.2. **Problem 67:** - The curve $ y = \frac{10}{x^2 - 2x - 24} $ bounds the region on one side. - The x-axis bounds the region from below. - Vertical lines at $ x = -2 $ and $ x = 2 $ bound the sides of the region.

Math

Calculus

67

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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#67

---
### Calculus: Areas of Regions

**66–68. Areas of Regions**
Find the area of the following regions.

\[ x - x^2 \]

---

**Explanation:**

In this exercise, you'll need to determine the area of the regions described by the given mathematical functions. The function you see, $ x - x^2 $, represents a parabola. Typically, for such problems, you will find the area under the curve between specified limits using integration.

To find the area of the region bounded by the curve $ y = x - x^2 $ and the x-axis, you can:

1. **Find the roots of the equation** $ x - x^2 = 0 $:
- Solve for $ x $:
$ x(x - 1) = 0 $
So, $ x = 0 $ or $ x = 1 $.

2. **Set up the definite integral**:
- The integral will be from 0 to 1 (the roots of the equation):
\[ \int_{0}^{1} (x - x^2) \, dx \]

3. **Evaluate the integral**:
- Perform the integration and simplify to find the area.

### Next Steps

To proceed, you should:
1. Solve the integral using standard integration techniques.
2. Verify your results by checking the points of intersection and ensuring the integral boundaries are correct.

These problems are fundamental in understanding how to calculate areas using integrals and are very common in calculus courses.

---

### Integral Calculus Problems for Area Determination

**Problem 66:** Determine the area of the region bounded by the curve $ y = \frac{x - x^2}{(x + 1)(x^2 + 1)} $ and the x-axis from $ x = 0 $ to $ x = 1 $.

**Problem 67:** Determine the area of the region bounded by the curve $ y = \frac{10}{x^2 - 2x - 24} $, the x-axis, and the vertical lines $ x = -2 $ and $ x = 2 $.

**Problem 68:** Determine the area of the region in the first quadrant bounded by the curves $ y = \frac{3x^2 + 2x + 1}{x(x^2 + x + 1)} $, $ y = \frac{2}{x} $, and the vertical line $ x = 2 $.

### Explanation of Mathematical Concepts and Methods
These problems involve finding the area of the regions bounded by given curves and specific lines or axes. To find these areas, one needs to set up definite integrals based on the given boundaries and the functions provided.

#### Steps to Solve:

1. **Identify the boundaries** of the area you need to find. This could involve particular functions, the x-axis, the y-axis, and vertical/horizontal lines.

2. **Set up the definite integral(s)** for the functions within the specified boundaries. This involves determining the points of intersection if necessary.

3. **Integrate the function(s)** within the limits to find the area.

### Detailed Explanation of the Graphs:
**Graphs Representation:**

1. **Problem 66:**
- The curve $ y = \frac{x - x^2}{(x + 1)(x^2 + 1)} $ bounds the region on one side.
- The x-axis bounds the region on another side from $ x = 0 $ to $ x = 1 $.

2. **Problem 67:**
- The curve $ y = \frac{10}{x^2 - 2x - 24} $ bounds the region on one side.
- The x-axis bounds the region from below.
- Vertical lines at $ x = -2 $ and $ x = 2 $ bound the sides of the region.

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