Expert ∫ Calculus Topic 93 of 104
Line Integrals
Integrate along a curve.
Key idea
Integrate along a curve.
In depth
∫_C F · dr adds up the component of F along the path, computing work done. Parametrise the curve, so ∫ F(r(t)) · r′(t) dt.
Worked example
F = ⟨y, x⟩ along r(t) = ⟨t, t²⟩, 0 ≤ t ≤ 1: ∫ (t²·1 + t·2t) dt = ∫ 3t² dt = 1.
Formula
∫ F·dr
Real-world uses
- Calculating the work done by a force moving along a wire or a path
Common mistake
Forgetting the r′(t) factor.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Line Integrals?
Integrate along a curve. It is a topic at the expert level in Calculus.
What is the formula for Line Integrals?
The key formula is ∫ F·dr.
Where is Line Integrals used in real life?
Real-world examples: Calculating the work done by a force moving along a wire or a path.
What should I learn before Line Integrals?
Make sure you are comfortable with Vector Fields, the previous topic in Calculus.
What is a common mistake with Line Integrals?
Forgetting the r′(t) factor.
How can I practise Line Integrals?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Line Integrals?
Move on to Conservative Fields next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.