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

Step by step

The question

Question: find the line integral of F = ⟨x, x⟩ along y = x² from (0, 0) to (1, 1).

F = ⟨x, x⟩

along y = x² from (0, 0) to (1, 1)

1 Write the path in t

On the curve, let x be t.

Then y is t².

t goes from 0 to 1.

x = t, y = t²

So x = t

x = t

and y = t², with t from 0 to 1.

y = t²

Then dx is the derivative of x, times dt,

dx = 1 dt

and dy the same for y.

dy = 2t dt

Here is the path on the grid.

It goes from (0, 0) to (1, 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 = t.

F = ⟨t, t⟩

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 1, plus t times 2t, all times dt.

P dx + Q dy = t × 1 dt + t × 2t dt

Multiply out and add the like terms: t + 2t².

P dx + Q dy = (t + 2t²) dt

Careful!

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

They carry the derivatives 1 and 2t.

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 (t + 2t²) dt = [t²2 + 2t³3] from 0 to 1

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

∫ from 0 to 1 of (t + 2t²) dt = ((1)²2 + 2(1)³3) − ((0)²2 + 2(0)³3)

The bottom part is 0.

So it is 7/6.

∫ from 0 to 1 of (t + 2t²) dt = 76

4 Check

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

About 1.17.

200 small steps: sum of F • step ≈ 1.167

Our answer is 1.167.

Almost the same.

Right!

7/6 = 1.167

Let's look back.

First: write the path in t.

Then: put the path in.

Then: integrate.

Last: check.

Answer