Work done by F = <y, 1> along the segment from (0, 0) to (3, 1) Step by Step

Step by step

The question

Question: find the work done by F = ⟨y, 1⟩ along the segment from (0, 0) to (3, 1).

F = ⟨y, 1⟩

along the segment from (0, 0) to (3, 1)

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 = 3t

x = 3t

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

y = t

Then dx is the derivative of x, times dt,

dx = 3 dt

and dy the same for y.

dy = 1 dt

Here is the path on the grid.

It goes from (0, 0) to (3, 1).

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 = 1.

F = ⟨t, 1⟩

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 3, plus 1 times 1, all times dt.

P dx + Q dy = t × 3 dt + 1 × 1 dt

Multiply out and add the like terms: 3t + 1.

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

Careful!

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

They carry the derivatives 3 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 (3t + 1) dt = [3t²2 + t] from 0 to 1

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

∫ from 0 to 1 of (3t + 1) dt = (3(1)²2 + (1)) − (3(0)²2 + (0))

The bottom part is 0.

So it is 5/2.

∫ from 0 to 1 of (3t + 1) dt = 52

4 Check

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

About 2.5.

200 small steps: sum of F • step ≈ 2.5

Our answer is 2.5.

Almost the same.

Right!

5/2 = 2.5

Let's look back.

First: write the path in t.

Then: put the path in.

Then: integrate.

Last: check.

Answer