Math Problem Statement
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. (Do this on paper. Your instructor may ask you to turn in this graph.) y=x^2- x, y=4 x+6 Then find the area S of the region.
Solution
To solve this problem, let's break it down step by step.
Step 1: Sketch the region enclosed by the curves and .
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Set up the equations:
- The first curve is a parabola: .
- The second curve is a line: .
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Find the points of intersection between the curves by setting the two equations equal: Simplify and solve the quadratic equation: Factor the quadratic: So, and .
These are the points of intersection: and .
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Plot the curves and shade the region between and . The region is bounded between these two points along the x-axis, with the parabola below the line.
Step 2: Decide whether to integrate with respect to or .
Since we are integrating between two vertical lines (from to ), we will integrate with respect to . The height of the approximating rectangle will be the vertical distance between the two curves, which is the difference in their -values at each .
Step 3: Draw a typical approximating rectangle.
- The width of the rectangle is a small change in , denoted by .
- The height of the rectangle is the difference between the two functions at any point :
Step 4: Set up the integral.
To find the area, we integrate the difference between the two functions over the interval to : Simplifying the integrand:
Step 5: Solve the integral.
First, find the indefinite integral:
Now, evaluate the definite integral:
Substitute the limits:
- At :
- At :
Now, subtract the values:
Thus, the area of the region is approximately 57.17 square units.
Step 6: Conclusion
- The area of the region enclosed by the curves is approximately 57.17 square units.
- The typical rectangle is vertical, with width and height .
Would you like further details or clarifications on any of the steps?
Here are some related questions to deepen your understanding:
- How would the process change if the curves were given in terms of instead of ?
- What would the integral look like if you had to integrate with respect to instead of ?
- How can you use the symmetry of the curves to simplify the integral?
- What happens to the area if the limits of integration are changed?
- How do the properties of parabolas and linear functions affect the shape of the enclosed region?
Tip: Always visualize the region before setting up the integral—it helps in deciding the limits of integration and choosing whether to integrate with respect to or .
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Math Problem Analysis
Mathematical Concepts
Integration
Area under Curves
Quadratic Equations
Linear Functions
Formulas
Definite Integral: A = ∫[a, b] f(x) dx
Difference of functions: Height = f(x) - g(x)
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12
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