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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Please answer the questions with steps and with correct answer
![**Problem:**
5. Shade in the area given by the integral
\[
\int_{1}^{2} y^2 \, dy.
\]
**Graph/Diagram Explanation:**
The diagram features a coordinate plane with axes labeled \( x \) and \( y \). The horizontal axis represents \( x \) and the vertical axis represents \( y \). The graph displays the curve of the function \( y = \sqrt{x} \), a standard square root function.
- **Axes and Origin:**
- The \( x \)-axis (horizontal) and \( y \)-axis (vertical) intersect at the origin \((0, 0)\).
- **Grid:**
- The background contains a grid composed of squares, providing reference points for plotting and reading coordinates.
- **Function \( y = \sqrt{x} \):**
- The graph shows this function beginning near the origin and increasing gradually. This curve represents the relationship where \( y \) is the square root of \( x \).
The task is to compute and shade the region corresponding to the definite integral \(\int_{1}^{2} y^2 \, dy\). This integral involves calculating the area between \( y = 1 \) and \( y = 2 \) on the \( y \)-axis, and \( y^2 \) as the function of \( y \).
**Note for Students:**
- The integral provided represents the area under the curve \( y^2 \) from \( y = 1 \) to \( y = 2 \), in the \( y \)-coordinate system, not along the path of \( y = \sqrt{x} \).
- Focus on understanding the transition between variables used in defining functions and calculating the area under those functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ba37d91-3081-42ea-952f-3d2c5579a63d%2F0540a306-d5fd-4eec-90f3-c6e00a0e9224%2Fi5rb63_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
5. Shade in the area given by the integral
\[
\int_{1}^{2} y^2 \, dy.
\]
**Graph/Diagram Explanation:**
The diagram features a coordinate plane with axes labeled \( x \) and \( y \). The horizontal axis represents \( x \) and the vertical axis represents \( y \). The graph displays the curve of the function \( y = \sqrt{x} \), a standard square root function.
- **Axes and Origin:**
- The \( x \)-axis (horizontal) and \( y \)-axis (vertical) intersect at the origin \((0, 0)\).
- **Grid:**
- The background contains a grid composed of squares, providing reference points for plotting and reading coordinates.
- **Function \( y = \sqrt{x} \):**
- The graph shows this function beginning near the origin and increasing gradually. This curve represents the relationship where \( y \) is the square root of \( x \).
The task is to compute and shade the region corresponding to the definite integral \(\int_{1}^{2} y^2 \, dy\). This integral involves calculating the area between \( y = 1 \) and \( y = 2 \) on the \( y \)-axis, and \( y^2 \) as the function of \( y \).
**Note for Students:**
- The integral provided represents the area under the curve \( y^2 \) from \( y = 1 \) to \( y = 2 \), in the \( y \)-coordinate system, not along the path of \( y = \sqrt{x} \).
- Focus on understanding the transition between variables used in defining functions and calculating the area under those functions.
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