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Problem 9(6pts). Let γ be the path tracing the graph of the function g(x)=sin(x) on the interval [0,π] (the path goes from (0,0) to (π,0) along the graph). Let ω=(cos(x)+y)dx+(4x+4y

3
)dy. Compute the integral ∫
γ

ω.

User Baldr
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1 Answer

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Answer:

To compute the integral ∫γω along the path γ, we need to parameterize the path γ and then evaluate the integral using the parameterization.

The path γ traces the graph of the function g(x) = sin(x) on the interval [0, π]. We can parameterize this path as follows:

x = t (where t ∈ [0, π])

y = sin(t)

Now, let's compute the differential forms dx and dy:

dx = dt

dy = cos(t) dt

Substituting these expressions into ω = (cos(x) + y) dx + (4x + 4y³) dy, we get:

ω = (cos(t) + sin(t)) dt + (4t + 4sin(t)³) cos(t) dt

= (cos(t) + sin(t) + (4t + 4sin(t)³) cos(t)) dt

Now, we can evaluate the integral along the path γ:

∫γ ω = ∫₀ᴨ (cos(t) + sin(t) + (4t + 4sin(t)³) cos(t)) dt

To evaluate this integral, we can split it into three separate integrals:

I₁ = ∫₀ᴨ cos(t) dt

I₂ = ∫₀ᴨ sin(t) dt

I₃ = ∫₀ᴨ (4t + 4sin(t)³) cos(t) dt

Evaluating each integral separately:

I₁ = [sin(t)]₀ᴨ = sin(ᴨ) - sin(0) = 0 - 0 = 0

I₂ = [-cos(t)]₀ᴨ = -cos(ᴨ) + cos(0) = -(-1) + 1 = 2

To evaluate I₃, we can use integration by parts:

Let u = 4t + 4sin(t)³ and dv = cos(t) dt

Then du = 4 + 12sin(t)² cos(t) dt and v = sin(t)

Using the formula for integration by parts, we have:

I₃ = uv - ∫v du

= (4t + 4sin(t)³) sin(t) - ∫sin(t) (4 + 12sin(t)² cos(t)) dt

= (4t + 4sin(t)³) sin(t) - ∫(4sin(t) + 12sin(t)³ cos(t)) dt

= (4t + 4sin(t)³) sin(t) - (4∫sin(t) dt + 12∫sin(t)³ cos(t) dt)

= (4t + 4sin(t)³) sin(t) - (-4cos(t) - 12I₃)

Simplifying the equation:

I₃ = (4t + 4sin(t)³) sin(t) + 4cos(t) + 12I₃

11I₃ = (4t + 4sin(t)³) sin(t) + 4cos(t)

I₃ = ((4t + 4sin(t)³) sin(t) + 4cos(t))/11

Now, substituting the values of I₁, I₂, and I₃ back into the original integral:

∫γ ω = I₁ + I₂ + I₃

= 0 + 2 + ((4t + 4sin(t)³) sin(t) + 4cos(t))/11

Therefore, the value of the integral ∫γω along the path γ is ((4t + 4sin(t)³) sin(t) + 4cos(t))/11 + 2.

Explanation:

User Aamin Khan
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