Math Problem Statement
ketch the region enclosed by x + y^2 = 20 and x + y = 20 . Decide whether to integrate with respect to x or y , and then find the area of the region.
The area is:
Solution
To solve the problem of finding the area of the region enclosed by the curves and , we can follow these steps:
1. Sketch the Curves
First, let's rewrite the equations to better understand the curves:
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Equation 1:
- Solving for :
- This represents a parabola opening to the left with vertex at .
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Equation 2:
- Solving for :
- This represents a straight line with a slope of .
2. Find the Points of Intersection
To find the points of intersection between the parabola and the line, set the equations equal to each other:
Simplifying, we get:
So, or .
Substituting these values of back into one of the original equations to find :
- For : .
- For : .
Thus, the points of intersection are and .
3. Decide the Method of Integration
Given the nature of the equations (especially the parabola opening horizontally), it is easier to integrate with respect to .
4. Set Up the Integral
The area can be found by integrating the difference between the two functions from to :
Simplifying the integrand:
5. Evaluate the Integral
Now, integrate:
Substituting the limits:
[ A = \left[-\frac{1^2}{2} + \frac{1^3}{3}\right] - \left[-\frac{0^2}{2} + \frac{0^3}{3}\right] ]
Since we are finding the area, we take the absolute value:
Final Answer:
The area of the region enclosed by the curves is .
Would you like more details on any part of this process, or do you have any questions?
Related Questions:
- How would you sketch the graph of ?
- What happens if you integrate with respect to instead?
- How do you find the intersection points of other types of curves?
- How does the direction in which a parabola opens affect the integration method?
- Can you explain the significance of taking the absolute value when calculating area?
Tip:
When deciding whether to integrate with respect to or , choose the variable that simplifies the integral and makes the bounds easier to determine.
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Math Problem Analysis
Mathematical Concepts
Integration
Area Calculation
Quadratic Equations
Formulas
Integration by Substitution
Theorems
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Suitable Grade Level
Grades 11-12