Line integral of F = <y, x> along the segment from (0, 0) to (1, 2) Step by Step

Step by step

The question

Question: find the line integral of F = ⟨y, x⟩ along the segment from (0, 0) to (1, 2).

F = ⟨y, x⟩

along the segment from (0, 0) to (1, 2)

1 Write the path in t

A segment: start, plus t times the step from start to end.

t goes from 0 to 1.

start + t × (end − start)

So x = t

x = t

and y = 2t, with t from 0 to 1.

y = 2t

Then dx is the derivative of x, times dt,

dx = 1 dt

and dy the same for y.

dy = 2 dt

Here is the path on the grid.

It goes from (0, 0) to (1, 2).

The blue arrows show F at a few points on the path.

2 Put the path in

Put x and y in t into F: P = 2t and Q = t.

F = ⟨2t, t⟩

The line integral is P dx plus Q dy, added along the path.

∫ P dx + Q dy

So P dx + Q dy is 2t times 1, plus t times 2, all times dt.

P dx + Q dy = 2t × 1 dt + t × 2 dt

Multiply out and add the like terms: 4t.

P dx + Q dy = 4t dt

Careful!

Many people put t into F but forget dx and dy.

They carry the derivatives 1 and 2.

Careful: dx and dy become dt pieces too

3 Integrate

Integrate each term: the power goes up by 1, and we divide by the new power.

∫ from 0 to 1 of 4t dt = 2t² from 0 to 1

Put in the top t = 1, then take away the bottom t = 0.

∫ from 0 to 1 of 4t dt = 2(1)² − 2(0)²

The bottom part is 0.

So it is 2.

∫ from 0 to 1 of 4t dt = 2

4 Check

Let's check another way.

Here ∂P/∂y equals ∂Q/∂x, so F has a potential: f = xy.

∂P/∂y = 1 = ∂Q/∂x: f = xy

Then the answer is f at the end minus f at the start: 2.

The same.

Right!

f(1, 2) − f(0, 0) = 2 − 0 = 2

Let's look back.

First: write the path in t.

Then: put the path in.

Then: integrate.

Last: check.

Answer