Math Problem Statement
- Find the area bounded by the curve x = cos t, y = e t , 0 ≤ t ≤ π 2 and the lines y = 1 and x = 0
Solution
To find the area bounded by the given parametric curve for , and the lines and , we need to set up an appropriate integral. We will follow these steps:
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Describe the region: The region is bounded by the curve , the vertical line , and the horizontal line .
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Find the parametric expressions for the area: The area under a parametric curve from to is given by: Here, and .
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Calculate :
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Set up the integral: The area is given by: Simplifying, this becomes:
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Evaluate the integral: We need to solve . We can use integration by parts, where we set and , leading to and .
Applying integration by parts:
Repeating integration by parts for , set and , leading to and : This results in: Combining terms gives: Therefore,
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Evaluate at the bounds: Calculating this, we get: Thus, the area is:
The area bounded by the curve and the given lines is .
Would you like more details or have any questions?
Here are five related questions:
- How would you find the area for different bounds of ?
- What if the parametric curve was given as a different function of ?
- How does integration by parts help in solving parametric integrals?
- What are other methods to find the area under a parametric curve?
- How do you handle cases where the curve intersects the boundary lines?
Tip: For parametric curves, always ensure to correctly set up the integral based on the derivatives of and .
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Math Problem Analysis
Mathematical Concepts
Parametric equations
Definite integrals
Integration by parts
Formulas
Area under parametric curve: A = ∫ g(t) (dx/dt) dt
Integration by parts: ∫ u dv = uv - ∫ v du
Theorems
Fundamental Theorem of Calculus
Integration by Parts
Suitable Grade Level
Grades 11-12 / Early College
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