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

Step by step

The question

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

F = ⟨−y, x⟩

along the segment from (1, 0) to (0, 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 = 1 − t

x = 1 − 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 (1, 0) to (0, 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 = 1 − t.

F = ⟨−2t, 1 − 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 (1 − t) times 2, all times dt.

P dx + Q dy = (−2t) × (−1) dt + (1 − t) × 2 dt

Multiply out and add the like terms: 2.

P dx + Q dy = 2 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 2 dt = 2t from 0 to 1

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

∫ from 0 to 1 of 2 dt = 2(1) − 2(0)

The bottom part is 0.

So it is 2.

∫ from 0 to 1 of 2 dt = 2

4 Check

Let's check: cut the path into 200 tiny steps and add F dot each step, by computer.

About 2.

200 small steps: sum of F • step ≈ 2

Our answer is 2.

Almost the same.

Right!

2 = 2

Let's look back.

First: write the path in t.

Then: put the path in.

Then: integrate.

Last: check.

Answer