Math Problem Statement
Solution
To find the area of the region enclosed by the line and the curve , let's go through the solution step by step.
Step 1: Set up the equations and find intersection points
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We have two functions:
- (a horizontal line)
- (a parabola opening upwards)
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To find the points of intersection, set these two equations equal to each other: Solving for : So, the points of intersection are at and .
Step 2: Set up the integral for the area
Since the line is above the curve within the interval , we can find the area between them by integrating the difference between the two functions from to .
Simplifying the integrand:
Step 3: Integrate
Now, integrate with respect to over the interval :
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Integrate :
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Integrate :
Step 4: Combine the results
The area is:
Final Answer:
The area of the region enclosed by the line and the curve is:
Would you like more details or have any questions?
Here are some additional questions related to this topic:
- How would the process change if the curve was ?
- What if the line was instead of ?
- How would you find the enclosed area if the parabola opened downward?
- Could you use symmetry to simplify the integration process in similar problems?
- How does changing the limits of integration affect the calculated area?
Tip: When finding the area between curves, always check which function is above the other over the given interval to set up the integral correctly.
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Math Problem Analysis
Mathematical Concepts
Integral Calculus
Area Between Curves
Formulas
Area = ∫(f(x) - g(x)) dx over [a, b]
Theorems
Fundamental Theorem of Calculus
Suitable Grade Level
Grades 11-12