Step by step
The question
Question: find the work done by F = ⟨x², y⟩ along y = 3x from (0, 0) to (1, 3).
1 Write the path in t
On the curve, let x be t.
Then y is 3t.
t goes from 0 to 1.
So x = t
and y = 3t, with t from 0 to 1.
Then dx is the derivative of x, times dt,
and dy the same for y.
Here is the path on the grid.
It goes from (0, 0) to (1, 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 = t² and Q = 3t.
The line integral is P dx plus Q dy, added along the path.
So P dx + Q dy is t² times 1, plus 3t times 3, all times dt.
Multiply out and add the like terms: t² + 9t.
Careful!
Many people put t into F but forget dx and dy.
They carry the derivatives 1 and 3.
3 Integrate
Integrate each term: the power goes up by 1, and we divide by the new power.
Put in the top t = 1, then take away the bottom t = 0.
The bottom part is 0.
So it is 29/6.
4 Check
Let's check another way.
Here ∂P/∂y equals ∂Q/∂x, so F has a potential: f = x³ over 3 + y² over 2.
Then the answer is f at the end minus f at the start: 29/6.
The same.
Right!
Let's look back.
First: write the path in t.
Then: put the path in.
Then: integrate.
Last: check.