Math Problem Statement
Solution
The problem asks to compute the line integral:
where is the line segment connecting points and .
Solution:
The line segment between and can be parametrized as:
This is because , , and for .
Now, we can calculate the differential line element . The magnitude of the derivative of is:
Now, substitute the values of , , and into the integrand:
Thus, the integral becomes:
The integral of is:
So, the result is:
Final Answer:
Let me know if you need further details or have any questions!
Here are 5 related questions to expand on this topic:
- How would the integral change if the line segment was not linear but a curve?
- Can you parametrize a different line integral between two arbitrary points?
- What is the physical interpretation of a line integral in this context?
- How do you compute a surface integral over a plane or curved surface?
- What is the difference between a line integral and a regular definite integral?
Tip: For line integrals, always ensure you correctly parametrize the path and compute the differential length accordingly.
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Math Problem Analysis
Mathematical Concepts
Vector Calculus
Line Integrals
Parametric Equations
Formulas
Line segment parametrization: r(t) = (t, t, t), where t ∈ [0, 1]
Line element: dl = √(1^2 + 1^2 + 1^2) dt = √3 dt
Integral calculation: ∫(x + y + z) dl
Theorems
Parametric representation of line integrals
Basic integration of polynomials
Suitable Grade Level
University - Calculus III
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