Line integral of F = <y, 2> along the segment from (0, 0) to (2, 3) Step by Step

Step by step

The question

Question: find the line integral of F = ⟨y, 2⟩ along the segment from (0, 0) to (2, 3).

F = ⟨y, 2⟩

along the segment from (0, 0) to (2, 3)

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 = 2t

x = 2t

and y = 3t, with t from 0 to 1.

y = 3t

Then dx is the derivative of x, times dt,

dx = 2 dt

and dy the same for y.

dy = 3 dt

Here is the path on the grid.

It goes from (0, 0) to (2, 3).

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 = 3t and Q = 2.

F = ⟨3t, 2⟩

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

∫ P dx + Q dy

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

P dx + Q dy = 3t × 2 dt + 2 × 3 dt

Multiply out and add the like terms: 6t + 6.

P dx + Q dy = (6t + 6) dt

Careful!

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

They carry the derivatives 2 and 3.

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 (6t + 6) dt = [3t² + 6t] from 0 to 1

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

∫ from 0 to 1 of (6t + 6) dt = (3(1)² + 6(1)) − (3(0)² + 6(0))

The bottom part is 0.

So it is 9.

∫ from 0 to 1 of (6t + 6) dt = 9

4 Check

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

About 9.

200 small steps: sum of F • step ≈ 9

Our answer is 9.

Almost the same.

Right!

9 = 9

Let's look back.

First: write the path in t.

Then: put the path in.

Then: integrate.

Last: check.

Answer