Sketch the region of integration. f(x, y) dx dy (0, 4) |(4, 4) (4, 4) y=x y=x X |(4, 0) y |(0, 4) |(0, 4) (4, 4) y=x y=x (4, 0) Change the order of integration. f(x, y) dy dx
Sketch the region of integration. f(x, y) dx dy (0, 4) |(4, 4) (4, 4) y=x y=x X |(4, 0) y |(0, 4) |(0, 4) (4, 4) y=x y=x (4, 0) Change the order of integration. f(x, y) dy dx
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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![## Region of Integration
Consider the double integral expressed in terms of \( dx \, dy \):
\[
\int_{0}^{4} \left( \int_{0}^{y} f(x, y) \, dx \right) dy
\]
This integral represents the area where we need to integrate the function \( f(x, y) \) over the specified region in the xy-plane.
### Step-by-Step Explanation and Graphs
1. **Initial Setup:**
- Given the integral limits, \( x \) ranges from 0 to \( y \) while \( y \) ranges from 0 to 4.
- The key line here is \( y = x \), which bounds the region.
2. **Graphical Representation:**
- Four diagrams show the sketching steps of the integration region.
#### Diagram 1:
- This first graph shows the line \( y = x \) in the first quadrant.
- Axes labeled: \( y \) on the vertical axis and \( x \) on the horizontal axis.
- Corners marked at (0,4), (4,4), and (4,0).
- The line \( y = x \) extends from the origin (0,0).
#### Diagram 2:
- Adds shading to the region between \( x = 0 \), \( y = 4 \), and \( y = x \).
- This shaded region is further delineated by marking points (0,4), (4,4), and the intercept at (4,0).
#### Diagram 3 and Diagram 4:
- Repeats shading of the area while clearly marking specific boundary points, helping visualize the integration limits.
- The diagonally shaded triangle re-emphasizes boundaries: \( y = x \), \( y = 4 \), and \( x = 4 \).
3. **Changing the Order of Integration:**
- For the integral transformation \( \int_{0}^{4} \left( \int_{x}^{4} f(x, y) \, dy \right) dx \):
- We are now integrating \( y \) with \( y \) limits from \( x \) to 4 and \( x \) limits from 0 to 4.
- Graphically, this does not alter the region's shape but reorients the bounds —](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b1222a1-b39c-4cb9-bb8f-961bd843d402%2F0a08b10f-5b24-4dbc-b30b-115b081779bf%2Fasjsvq_processed.png&w=3840&q=75)
Transcribed Image Text:## Region of Integration
Consider the double integral expressed in terms of \( dx \, dy \):
\[
\int_{0}^{4} \left( \int_{0}^{y} f(x, y) \, dx \right) dy
\]
This integral represents the area where we need to integrate the function \( f(x, y) \) over the specified region in the xy-plane.
### Step-by-Step Explanation and Graphs
1. **Initial Setup:**
- Given the integral limits, \( x \) ranges from 0 to \( y \) while \( y \) ranges from 0 to 4.
- The key line here is \( y = x \), which bounds the region.
2. **Graphical Representation:**
- Four diagrams show the sketching steps of the integration region.
#### Diagram 1:
- This first graph shows the line \( y = x \) in the first quadrant.
- Axes labeled: \( y \) on the vertical axis and \( x \) on the horizontal axis.
- Corners marked at (0,4), (4,4), and (4,0).
- The line \( y = x \) extends from the origin (0,0).
#### Diagram 2:
- Adds shading to the region between \( x = 0 \), \( y = 4 \), and \( y = x \).
- This shaded region is further delineated by marking points (0,4), (4,4), and the intercept at (4,0).
#### Diagram 3 and Diagram 4:
- Repeats shading of the area while clearly marking specific boundary points, helping visualize the integration limits.
- The diagonally shaded triangle re-emphasizes boundaries: \( y = x \), \( y = 4 \), and \( x = 4 \).
3. **Changing the Order of Integration:**
- For the integral transformation \( \int_{0}^{4} \left( \int_{x}^{4} f(x, y) \, dy \right) dx \):
- We are now integrating \( y \) with \( y \) limits from \( x \) to 4 and \( x \) limits from 0 to 4.
- Graphically, this does not alter the region's shape but reorients the bounds —
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