Line integral of F = <y^2, x> along the segment from (0, 0) to (1, 3) Step by Step

Step by step

The question

Question: find the line integral of F = ⟨y², x⟩ along the segment from (0, 0) to (1, 3).

F = ⟨y², x⟩

along the segment from (0, 0) to (1, 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 = t

x = t

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

y = 3t

Then dx is the derivative of x, times dt,

dx = 1 dt

and dy the same for y.

dy = 3 dt

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 = 9t² and Q = t.

F = ⟨9t², t⟩

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

∫ P dx + Q dy

So P dx + Q dy is 9t² times 1, plus t times 3, all times dt.

P dx + Q dy = 9t² × 1 dt + t × 3 dt

Multiply out and add the like terms: 9t² + 3t.

P dx + Q dy = (9t² + 3t) dt

Careful!

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

They carry the derivatives 1 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 (9t² + 3t) dt = [3t³ + 3t²2] from 0 to 1

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

∫ from 0 to 1 of (9t² + 3t) dt = (3(1)³ + 3(1)²2) − (3(0)³ + 3(0)²2)

The bottom part is 0.

So it is 9/2.

∫ from 0 to 1 of (9t² + 3t) 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