Line integrals are a mathematical construct used to estimate quantities such as work done by a force on a curved path or the flow field along a curve.

In this article on line integrals, we will explore what line integrals are, their types, and how to compute them. We will also provide solved examples and practice questions to help you understand and master the concept of line integrals.

What are Line Integrals?

Line integrals are used for the integration of a function along a curve in a plane or space. They help in finding such quantities as work done by fields of force, or flux of vector fields. Line integrals can be calculated with respect to arc length or with respect to the coordinate variables. It means parameterizing the curve and expressing the function in terms of the parameterization, then integrating over the range of the parameter.

Types of Line Integrals

1. Line Integral of Scalar Fields

A line integral of a scalar field involves integrating a scalar function along a curve. It is used to find quantities such as length, mass, and work. The general formula for a line integral of a scalar field is:

[Tex]∫ C ​ f(x,y)ds[/Tex]

Here, (ds) represents an infinitesimal segment of the curve C, and f(x, y) is the scalar function being integrated.

2. Line Integral of Vector Fields

A line integral of a vector field involves integrating a vector field along a curve. It is commonly used in physics to calculate work done by a force field along a path. The general formula for a line integral of a vector field is:

[Tex]∫ C ​ F⋅dr[/Tex]

Here, F is the vector field and dr represents an infinitesimal displacement vector along the curve C.

How to Compute Line Integrals

Parameterizing the Curve

To compute a line integral, the curve C must be parameterized. This involves expressing the coordinates x and y as functions of a parameter t:

[Tex]x=x(t),y=y(t),a≤t≤b[/Tex]

This parameterization converts the integral over the curve into an integral over the parameter t.

Line Integral Formula of Scalar Fields

Line Integral Formula of Vector Fields

Line integrals provide a basic tool for understanding and quantifying the process of change along paths in space, making them of central importance in a subject like Electromagnetism and Fluid Dynamics.

Line Integrals Practice Questions – Solved

These solved Line Integrals Practice Questions will help you understand the application of the concept and how to use it to solve various problems.

1: Evaluate ∫_C(x + y) ds, where C is the straight line from (0, 0) to (1, 1).

Parameterize the line: x = t, y = t, 0 ≤ t ≤ 1

ds = √((dx/dt)² + (dy/dt)²) dt = √2 dt

∫C (x + y) ds = ∫0¹ (t + t) √2 dt

= 2√2 ∫₀¹ t dt

= 2√2 [t²/2]₀¹ = √2

2: Evaluate ∫C F · dr, where F(x, y) = yi + xj and C is the unit circle centred at the origin, traversed counterclockwise.

We are going to parameterize the circle i.e. x = cos(t), y = sin(t), 0 ≤ t ≤ 2π

dr = (-sin(t)i + cos(t)j) dt

F · dr = (sin(t)cos(t) + cos(t)sin(t)) dt = sin(2t) dt

∫_C F · dr = ∫₀²π sin(2t) dt

= π[-cos(2t)/2]₀² = 0

3: Calculate the work done by the force field F(x, y) = (2x+y)i + (x-y)j along the path from (0, 0) to (2, 1) following the parabola y = x²/4.

In this problem, we will parameterize the path: x = t, y = t²/4, 0 ≤ t ≤ 2

dr = (1)i + (t/2)j dt

F · dr = (2t + t²/4) + (t – t²/4)(t/2) dt = (2t + t²/4 + t²/2 – t³/8) dt

∫_C F · dr = ∫0² (2t + 3t²/4 – t³/8) dt

= [t² + t³/4 – t⁴/32]₀²

= 4 + 2 – 1 = 5

4: Find the circulation of F(x,y) = yi – xj around the square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1), traversed counterclockwise.

We are initially going to break the path into 4 line segments:

C1: y = 0, 0 ≤ x ≤ 1

C2: x = 1, 0 ≤ y ≤ 1

C3: y = 1, 1 ≥ x ≥ 0

C4: x = 0, 1 ≥ y ≥ 0

Now, we evaluate each segment:

∫_C1 F · dr = ∫₀¹ 0 dx = 0

∫_C2 F · dr = ∫₀¹ y dy = [y²/2]₀¹ = 1/2

∫_C3 F · dr = ∫₁⁰ (-1) dx = 1

∫_C4 F · dr = ∫₁⁰ (-y) dy = [-y²/2]₁⁰ = 1/2

Hence, Sum the results will be: 0 + 1/2 + 1 + 1/2 = 2

5: Calculate the flux of F(x, y) = xi + yj across the curve y = x², from x = 0 to x = 1.

We have parameterize the curve i.e. x = t, y = t², 0 ≤ t ≤ 1

Normal vector n = (-2t)i + j

F · n = -2t² + t²

Flux = ∫_C F · n ds = ∫₀¹ (-t²) √(1 + 4t²) dt

This integral is difficult to evaluate analytically, so we need to use the numerical methods to approximate it.

6: Show that the line integral of F(x,y) = (2x+y)i + (x+2y)j is independent of path by evaluating it along two different paths from (0,0) to (1,1).

For straight line y = x

We are going to parameterize i.e. x = t, y = t, 0 ≤ t ≤ 1

dr = i + j dt

F · dr = (2t + t) + (t + 2t) dt = 6t dt

∫_C F · dr = ∫0¹ 6t dt = [3t²]₀¹ = 3

Line Integrals Practice Questions- Unsolved

Solve these Line Integrals Practice Questions to test your understanding on the concept of Line Integrals.

1. Evaluate ∫_C (x + y) ds, where C is the line segment from (0,0) to (1,1).

2. Calculate ∫_C (x² + y²) ds, where C is the circle x² + y² = 4 traversed counterclockwise.

3. Find the work done by the force field F(x,y) = (y, -x) along the path from (1,0) to (0,1) along the parabola y = x².

4. Compute ∫_C (xy dx + x² dy), where C is the curve y = x³ from (0,0) to (1,1).

5. Evaluate ∫_C (e^x cos y dx + e^x sin y dy), where C is the straight line from (0,0) to (ln 2, π/4).

6. Calculate the line integral of f(x,y,z) = xyz along the helix r(t) = (cos t, sin t, t) for 0 ≤ t ≤ 2π.

7. Find ∫_C (x dy – y dx), where C is the ellipse x²/4 + y²/9 = 1 traversed clockwise.

8. Evaluate ∫_C (y dx – x dy) / (x² + y²), where C is the upper half of the circle x² + y² = 1.

9. Compute the work done by F(x,y) = (2xy, x²) along the path from (0,1) to (2,3) following y = x + 1.

10. Calculate ∫_C (z dx + x dy + y dz), where C is the intersection of the plane x + y + z = 1 and the cylinder x² + y² = 1.

Read More,

• Integration by Parts
• Integration by Substitution Formula
• Integration by Partial Fractions

Line Integrals Practice Questions – FAQs

What is a Line Integral?

A line integral is a type of integral that evaluates a function along a curve or path in a plane or space. It’s used to calculate quantities such as work done by a force or the circulation of a vector field.

What are Two Main Types of Line Integrals?

Two main types of line integrals are:

• Line integrals of scalar fields (integrating a scalar function along a path)
• Line integrals of vector fields (integrating a vector function along a path)

What is Notation for a Line Integral?

A line integral is typically denoted as ∫_C f(x,y) ds or ∫_C F · dr, where C is the path of integration, f(x, y) is a scalar function, F is a vector field, and ds or dr represents an infinitesimal segment of the path.

What are Applications of Line Integrals?

Line integrals are used in physics to calculate work done by a force, in fluid dynamics to determine flow rates, in electromagnetics to compute potential differences, and in various areas of mathematics and engineering.

What is Green’s Theorem and How does it relate to Line Integrals?

Green’s theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by the curve. It’s used to convert between line integrals and area integrals in two dimensions.

How do you Evaluate a Line Integral?

To evaluate a line integral, we can use the following steps:

• Parameterize the path of integration
• Express the integrand in terms of the parameter
• Determine the limits of integration
• Perform the resulting single-variable integral