In order to use the fundamental theorem of line integrals, you need to find a scalar potential function - that is, a scalar function f(x, y) for which

grad f(x, y) = F(x, y)

This amounts to solving for f such that

∂f/dx = x + 3

∂f/∂y = 6y + 3

Integrating both sides of the first equation with respect to x gives

f = 1/2 x ^2 + 3x + g(y)

Differentiating with respect to y gives

∂f/∂y = dg/dy = 6y + 3

Solving for g gives

g = ∫ (6y + 3) dy = 3y ^2 + 3y + C

and hence

f(x, y) = 1/2 x ^2 + 3x + 3y ^2 + 3y + C

(a) By the fundamental theorem, the integral of F along any path starting at the point P (1, 0) and ending at Q (3, 3) is

∫ F(x, y) • dr = f (3, 3) - f (1, 0) = 99/2 - 7/2 = 46

(b) Now we're talking about a closed path, so the integral is simply 0. We can verify this by checking the integral over the origin-containing paths:

• From the origin to P :

∫ F(x, y) • dr = f (1, 0) - f (0, 0) = 7/2 - 0 = 7/2

• From Q back to the origin:

∫ F(x, y) • dr = f (0, 0) - f (3, 3) = 0 - 99/2 = -99/2

Then the total integral is 7/2 + 46 - 99/2 = 0, as expected.

6 0

3 years ago

Is y/7 = 2/x direct variation?

jenyasd209 [6]

Answer:

It is not direct variation

Step-by-step explanation:

y/7 = 2/x

y = 7×2/x

y = 14/x

And

y = k/x

k = 14 which is the constant of proportionality

: y ∞ 1/x

This is inverse variation

3 0

3 years ago

For a recent 41-year period, there were 5,469 major earthquakes; these are earthquakes measuring at least 6.0 on the Richter sca

pychu [463]

Below is the formula:

Probability Formula

Probability of an Event = Number of Favorable Outcomes / Total Number of Possible Outcomes

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5 0

3 years ago

What is this answer?

sergij07 [2.7K]

A because they all something I forgot but a

6 0

3 years ago