Textbook Question
10 10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
k = 1 k = 1
10
c. Σ (aₖ + bₖ - 1)
k = 1
1
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Ch. 5 - Integrals
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 5, Problem 5.PE.27
Find the area of the “triangular” region bounded on the left by x + y = 2, on the right by y = x², and above by y = 2.
Verified step by step guidance1
First, identify the region bounded by the three curves: the line \( x + y = 2 \), the parabola \( y = x^2 \), and the horizontal line \( y = 2 \). Sketching these curves can help visualize the triangular region.
Rewrite the line equation \( x + y = 2 \) in terms of \( y \) to get \( y = 2 - x \). This will help in setting up the integral.
Determine the points of intersection between the curves to find the limits of integration. Specifically, find where \( y = 2 - x \) intersects \( y = x^2 \), and where these curves intersect \( y = 2 \).
Decide whether to integrate with respect to \( x \) or \( y \). Since the region is bounded vertically by \( y = 2 \) and horizontally by curves expressed as functions of \( x \), integrating with respect to \( y \) might be simpler.
Set up the integral for the area by expressing the horizontal distance between the left and right boundaries as a function of \( y \). For each \( y \) between the lower and upper bounds, find the corresponding \( x \)-values on the line and parabola, then integrate the difference \( x_{right} - x_{left} \) with respect to \( y \) over the appropriate interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Finding Points of Intersection
To determine the boundaries of the region, it is essential to find where the given curves intersect. This involves solving equations simultaneously, such as setting y = 2 equal to the other curves or equating x + y = 2 and y = x² to find intersection points that define the limits of integration.
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Critical Points
Setting Up Definite Integrals for Area
The area between curves can be found by integrating the difference of the functions over the interval defined by their intersection points. Identifying which curve is on top or right and which is on bottom or left is crucial to correctly set up the integral representing the area.
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Understanding Region Boundaries in the Plane
Interpreting the problem requires visualizing or sketching the region bounded by the given lines and curves. Recognizing that the region is enclosed by a line, a parabola, and a horizontal line helps in deciding the order of integration and the limits, ensuring the correct calculation of the area.
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Related Practice