Mathematics · University
Conservative-field line integral
This applies when the vector field is the gradient of a single-valued potential on the path.
Method and conditions
This applies when the vector field is the gradient of a single-valued potential on the path.
∫C ∇φ · dr = φ(B) − φ(A)
Worked examples
Let F = ∇φ with φ(x,y) = 1x² + 6y². Find the line integral of F from (0,0) to (3,9). Round the final result to 4 decimal places if needed.
The fundamental theorem for line integrals gives φ(3,9) − φ(0,0).
495
Let F = ∇φ with φ(x,y) = 2x² + 5y². Find the line integral of F from (0,0) to (4,8). Round the final result to 4 decimal places if needed.
The fundamental theorem for line integrals gives φ(4,8) − φ(0,0).
352
Try it
Let F = ∇φ with φ(x,y) = 1x² + 6y². Find the line integral of F from (0,0) to (8,1). Round the final result to 4 decimal places if needed.
Let F = ∇φ with φ(x,y) = 2x² + 8y². Find the line integral of F from (0,0) to (6,2). Round the final result to 4 decimal places if needed.
Check your work
Answers and marking notes
- 70
The fundamental theorem for line integrals gives φ(8,1) − φ(0,0).
- 104
The fundamental theorem for line integrals gives φ(6,2) − φ(0,0).
Further reading and review references
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