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

Step by step

The question

Question: find the line integral of F = ⟨x − y, x + y⟩ along the segment from (1, 1) to (3, 2).

F = ⟨x − y, x + y⟩

along the segment from (1, 1) to (3, 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 + 2t

x = 1 + 2t

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

y = 1 + t

Then dx is the derivative of x, times dt,

dx = 2 dt

and dy the same for y.

dy = 1 dt

Here is the path on the grid.

It goes from (1, 1) to (3, 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 = t and Q = 2 + 3t.

F = ⟨t, 2 + 3t⟩

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

∫ P dx + Q dy

So P dx + Q dy is t times 2, plus (2 + 3t) times 1, all times dt.

P dx + Q dy = t × 2 dt + (2 + 3t) × 1 dt

Multiply out and add the like terms: 5t + 2.

P dx + Q dy = (5t + 2) dt

Careful!

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

They carry the derivatives 2 and 1.

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 (5t + 2) dt = [5t²2 + 2t] from 0 to 1

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

∫ from 0 to 1 of (5t + 2) dt = (5(1)²2 + 2(1)) − (5(0)²2 + 2(0))

The bottom part is 0.

So it is 9/2.

∫ from 0 to 1 of (5t + 2) dt = 92

4 Check

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

About 4.5.

200 small steps: sum of F • step ≈ 4.5

Our answer is 4.5.

Almost the same.

Right!

9/2 = 4.5

Let's look back.

First: write the path in t.

Then: put the path in.

Then: integrate.

Last: check.

Answer