What is the area of the region in the first quadrant bounded above by y = -2x + 8 and below by y = x²₁ IX. 4 3 543-2 -11° 1 2 3 4 5 2233 colm O O O 28 16
What is the area of the region in the first quadrant bounded above by y = -2x + 8 and below by y = x²₁ IX. 4 3 543-2 -11° 1 2 3 4 5 2233 colm O O O 28 16
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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![### Area of a Region in the First Quadrant
**Question:**
What is the area of the region in the first quadrant bounded above by \( y = -2x + 8 \) and below by \( y = x^2 \)?
#### Diagram Description:
The diagram below shows the region of interest:
- The x-axis ranges from \(-5\) to \(5\).
- The y-axis ranges from \(0\) to \(9\).
- The line \( y = -2x + 8 \) intersects the y-axis at \( y = 8 \) and declines with a slope of \(-2\).
- The curve \( y = x^2 \) represents a parabola opening upwards, intersecting the x-axis at \( (0, 0) \).
- The region of interest, shaded in the first quadrant, is bounded by these two equations where they intersect and form a closed area.
#### Options:
- \( \frac{32}{3} \)
- \( \frac{28}{3} \)
- \( \frac{16}{3} \)
- \( \frac{8}{3} \)
To solve this, consider finding the points of intersection between the line \( y = -2x + 8 \) and the curve \( y = x^2 \) by setting the equations equal to each other:
\[ -2x + 8 = x^2 \]
Solve for \( x \) to find the limits of integration along the x-axis and integrate the difference between the functions \(-2x + 8\) and \(x^2\) from the lower to the upper intersection points. The exact procedure involves determining these points algebraically and performing the actual definite integral computations to find the area of the shaded region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6539f692-5656-46fd-b828-f0fe9bb331a2%2F831ac871-f576-4b28-9548-611bdeb90e2a%2F3z8r7rxs_processed.png&w=3840&q=75)
Transcribed Image Text:### Area of a Region in the First Quadrant
**Question:**
What is the area of the region in the first quadrant bounded above by \( y = -2x + 8 \) and below by \( y = x^2 \)?
#### Diagram Description:
The diagram below shows the region of interest:
- The x-axis ranges from \(-5\) to \(5\).
- The y-axis ranges from \(0\) to \(9\).
- The line \( y = -2x + 8 \) intersects the y-axis at \( y = 8 \) and declines with a slope of \(-2\).
- The curve \( y = x^2 \) represents a parabola opening upwards, intersecting the x-axis at \( (0, 0) \).
- The region of interest, shaded in the first quadrant, is bounded by these two equations where they intersect and form a closed area.
#### Options:
- \( \frac{32}{3} \)
- \( \frac{28}{3} \)
- \( \frac{16}{3} \)
- \( \frac{8}{3} \)
To solve this, consider finding the points of intersection between the line \( y = -2x + 8 \) and the curve \( y = x^2 \) by setting the equations equal to each other:
\[ -2x + 8 = x^2 \]
Solve for \( x \) to find the limits of integration along the x-axis and integrate the difference between the functions \(-2x + 8\) and \(x^2\) from the lower to the upper intersection points. The exact procedure involves determining these points algebraically and performing the actual definite integral computations to find the area of the shaded region.
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