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Consider the following equations. 1 x² O 6 Sketch the region bounded by the graphs of the equations. y 5 3 2 1 6 5 3 2 y = 1 1 1 2 y = 4, 2 3 X = 6 3 4 4 5 5 6 6 7 7 X X

Consider the following equations. 1 x² O 6 Sketch the region bounded by the graphs of the equations. y 5 3 2 1 6 5 3 2 y = 1 1 1 2 y = 4, 2 3 X = 6 3 4 4 5 5 6 6 7 7 X X

Calculus: Early Transcendentals
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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It's only 1 question I put 2 images because it wont fit in one

### Integral Area Calculation Exercise Below are two graphs depicting regions under curves which are bounded by vertical lines, and shaded areas which represent the regions whose areas you need to evaluate. **Graph 1 Description:** - **Axis Labels:** The horizontal axis is labeled \(x\) and the vertical axis is labeled \(y\). - **Curve:** There is a black curve within the first quadrant that decreases steeply as \(x\) increases, starting at a high value near the y-axis and approaching zero. - **Boundaries:** - Left: \(x = 1\) - Right: \(x = 4\) - Horizontal line at \(y = 0\) - The curve itself from \(x = 1\) to \(x = 4\) - **Shaded Area:** The region under the curve between \(x = 1\) and \(x = 4\). **Graph 2 Description:** - **Axis Labels:** The horizontal axis is labeled \(x\) and the vertical axis is labeled \(y\). - **Curve:** There is a black curve within the first quadrant that initially decreases as \(x\) increases, starting higher near the y-axis and gradually reducing. - **Boundaries:** - Left: \(x = 1\) - Right: \(x = 4\) - Horizontal line at \(y = 6\) - The curve itself from \(x = 1\) to \(x = 4\) - **Shaded Area:** The region between the curve and the horizontal line at \(y = 6\) from \(x = 1\) to \(x = 4\). **Task:** Find the area of each shaded region. **Input Box:** There's a field labeled “Find the area of the region” where you would insert the computed area following your analysis. To determine the areas of these regions, you will need to set up and evaluate definite integrals based on the functions of the curves and the given boundaries.
Transcribed Image Text:### Integral Area Calculation Exercise Below are two graphs depicting regions under curves which are bounded by vertical lines, and shaded areas which represent the regions whose areas you need to evaluate. **Graph 1 Description:** - **Axis Labels:** The horizontal axis is labeled \(x\) and the vertical axis is labeled \(y\). - **Curve:** There is a black curve within the first quadrant that decreases steeply as \(x\) increases, starting at a high value near the y-axis and approaching zero. - **Boundaries:** - Left: \(x = 1\) - Right: \(x = 4\) - Horizontal line at \(y = 0\) - The curve itself from \(x = 1\) to \(x = 4\) - **Shaded Area:** The region under the curve between \(x = 1\) and \(x = 4\). **Graph 2 Description:** - **Axis Labels:** The horizontal axis is labeled \(x\) and the vertical axis is labeled \(y\). - **Curve:** There is a black curve within the first quadrant that initially decreases as \(x\) increases, starting higher near the y-axis and gradually reducing. - **Boundaries:** - Left: \(x = 1\) - Right: \(x = 4\) - Horizontal line at \(y = 6\) - The curve itself from \(x = 1\) to \(x = 4\) - **Shaded Area:** The region between the curve and the horizontal line at \(y = 6\) from \(x = 1\) to \(x = 4\). **Task:** Find the area of each shaded region. **Input Box:** There's a field labeled “Find the area of the region” where you would insert the computed area following your analysis. To determine the areas of these regions, you will need to set up and evaluate definite integrals based on the functions of the curves and the given boundaries.
## Bounded Region by Graphs of Equations Consider the following equations: \[ y = \frac{1}{x^2}, \quad y = 4, \quad x = 6 \] Sketch the region bounded by the graphs of the equations. ### Graph 1 The first graph displays the curves and lines: - \( y = \frac{1}{x^2} \): This is a decreasing curve that starts high on the y-axis when x is close to zero and asymptotically approaches the x-axis as x increases. This curve is a classic example of an inverse square relationship. - \( y = 4 \): This is a horizontal line cutting the y-axis at y = 4. - \( x = 6 \): This is a vertical line crossing the x-axis at x = 6. The shaded region in this graph is the area under the curve \( y = \frac{1}{x^2} \), above the line \( y = 4 \), and to the left of the line \( x = 6 \). This area is visually represented in blue. ### Graph 2 The second graph also displays the curves and lines: - \( y = \frac{1}{x^2} \): The same inverse square curve as described above. - \( y = 4 \): The same horizontal line as described above. - \( x = 6 \): The same vertical line as described above. The shaded region in this graph is the area under the curve \( y = \frac{1}{x^2} \), above the line \( y = 4 \), but focuses more closely on the behavior of the curve near the y-axis. This area is also visually represented in blue. These graphs facilitate the understanding of the bounded region determined by the given equations. The visual representation helps to understand the limits and areas under the curve and between the lines.
Transcribed Image Text:## Bounded Region by Graphs of Equations Consider the following equations: \[ y = \frac{1}{x^2}, \quad y = 4, \quad x = 6 \] Sketch the region bounded by the graphs of the equations. ### Graph 1 The first graph displays the curves and lines: - \( y = \frac{1}{x^2} \): This is a decreasing curve that starts high on the y-axis when x is close to zero and asymptotically approaches the x-axis as x increases. This curve is a classic example of an inverse square relationship. - \( y = 4 \): This is a horizontal line cutting the y-axis at y = 4. - \( x = 6 \): This is a vertical line crossing the x-axis at x = 6. The shaded region in this graph is the area under the curve \( y = \frac{1}{x^2} \), above the line \( y = 4 \), and to the left of the line \( x = 6 \). This area is visually represented in blue. ### Graph 2 The second graph also displays the curves and lines: - \( y = \frac{1}{x^2} \): The same inverse square curve as described above. - \( y = 4 \): The same horizontal line as described above. - \( x = 6 \): The same vertical line as described above. The shaded region in this graph is the area under the curve \( y = \frac{1}{x^2} \), above the line \( y = 4 \), but focuses more closely on the behavior of the curve near the y-axis. This area is also visually represented in blue. These graphs facilitate the understanding of the bounded region determined by the given equations. The visual representation helps to understand the limits and areas under the curve and between the lines.
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