Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. y = 2x + 2, y = 11-x², x = -1, x = 2 y -10 -10 -5 Find the area of the region. -10 y 10 5 5 10 X X 10 -10 -10 -5 y 10 5 A 5 -10 10 10 X X
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. y = 2x + 2, y = 11-x², x = -1, x = 2 y -10 -10 -5 Find the area of the region. -10 y 10 5 5 10 X X 10 -10 -10 -5 y 10 5 A 5 -10 10 10 X X
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. y = 2x + 2, y = 11-x², x = -1, x = 2 y -10 -10 -5 Find the area of the region. -10 y 10 5 5 10 X X 10 -10 -10 -5 y 10 5 A 5 -10 10 10 X X
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle.
please explain how each line of the curve was obtained and show in calculation
Transcribed Image Text:### Understanding the Problem of Finding the Area Enclosed by Curves
In this educational task, we explore how to find the area enclosed by different curves given by:
- \( y = 2x + 2 \) (a linear equation),
- \( y = 11 - x^2 \) (a quadratic equation),
- Vertical lines \( x = -1 \) and \( x = 2 \).
#### Graphical Analysis:
There are four graphs that exhibit the region enclosed by these curves. Each graph shows how the functions intersect and form a bounded area.
1. **Linear and Quadratic Functions:**
- The graph of \( y = 2x + 2 \) is a straight line with a positive slope intersecting the y-axis at 2.
- The graph of \( y = 11 - x^2 \) is an upside-down parabola with a vertex at (0, 11).
2. **Bounding Vertical Lines:**
- The vertical line \( x = -1 \) serves as the left boundary.
- The vertical line \( x = 2 \) acts as the right boundary.
3. **Shaded Region:**
- The enclosed area is shaded in blue in each graph.
4. **Approximating Rectangle:**
- A typical approximating rectangle, which is essential for visualizing integration, is shown within the shaded region. It is oriented vertically in this case, indicating integration with respect to \( x \).
#### Problem Objective:
Determine which graph accurately represents the solution based on integration features and correctly sketch the typical rectangle. Also, calculate the exact area of the shaded region.
#### Conclusion:
To find the area, integrate with respect to \( x \) from \( x = -1 \) to \( x = 2 \), using the functions \( y = 11 - x^2 \) and \( y = 2x + 2 \). Set up the integral for the difference of the two curves and solve to compute the area.
This exercise strengthens the understanding of visualizing integrals and solving for areas between curves.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Expert Solution
Step 1: Analysis and Introduction
Given information:
Curves:
Vertical Lines:
To find:
The area of the surface enclosed by the provided curves and lines.
Concept used:
Let the region be enclosed by which is bounded below by in the interval .
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