Next Question In the figure, the equation of the solid parabola is y=x² -2 and the equation of the dashed line is y =x. Determine the area of the shaded region.
Next Question In the figure, the equation of the solid parabola is y=x² -2 and the equation of the dashed line is y =x. Determine the area of the shaded region.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Problem Statement
In the figure, the equation of the solid parabola is \( y = x^2 - 2 \) and the equation of the dashed line is \( y = x \). Determine the area of the shaded region.
### Diagram Explanation
The image contains a coordinate system with a solid red parabola and a dashed blue line. The solid parabola, representing the function \( y = x^2 - 2 \), is a standard parabola that opens upwards and is shifted down by 2 units on the y-axis. The dashed line represents the equation \( y = x \), which is a straight line that passes through the origin with a slope of 1.
The two curves intersect, creating a shaded region between them. This region is bounded on the left by the intersection of the curves and on the right extends towards the positive x-direction.
### Task
To find the exact area of the shaded region, calculate the definite integral of the difference between the functions \( y = x \) and \( y = x^2 - 2 \) over the interval determined by their points of intersection.
**The area of the shaded region is:**
\[ \boxed{\ } \] (Type an exact answer.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb96afbd2-8574-4120-ad2b-59a4e6d0f8c2%2F89af420e-c4f5-443e-98b3-e563695ec26d%2Fr3lmhwv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
In the figure, the equation of the solid parabola is \( y = x^2 - 2 \) and the equation of the dashed line is \( y = x \). Determine the area of the shaded region.
### Diagram Explanation
The image contains a coordinate system with a solid red parabola and a dashed blue line. The solid parabola, representing the function \( y = x^2 - 2 \), is a standard parabola that opens upwards and is shifted down by 2 units on the y-axis. The dashed line represents the equation \( y = x \), which is a straight line that passes through the origin with a slope of 1.
The two curves intersect, creating a shaded region between them. This region is bounded on the left by the intersection of the curves and on the right extends towards the positive x-direction.
### Task
To find the exact area of the shaded region, calculate the definite integral of the difference between the functions \( y = x \) and \( y = x^2 - 2 \) over the interval determined by their points of intersection.
**The area of the shaded region is:**
\[ \boxed{\ } \] (Type an exact answer.)
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