Question

3. evaluate ∫C

​G(x,y)dx,∫C
​
G(x,y)dy and ∫C
​
G(x,y)ds on the indicated curue C. G(x,y)=3x^2
+6y^2;y=2x+1,−1≤x≤0

Answer 1

To evaluate the integrals on the curve C, we need to find the parametric equations of the curve C and substitute them into the function G(x,y). Then, we can calculate each integral using the appropriate formulas.

Step-by-step explanation:

To evaluate the integrals ∫C G(x,y)dx, ∫C G(x,y)dy, and ∫C G(x,y)ds on the curve C, we need to find the parametric equations of the curve C.

Given that y = 2x + 1 and -1 ≤ x ≤ 0, we can express x in terms of y as x = (y - 1) / 2.

Substituting this into G(x,y) = 3x^2 + 6y^2, we get G(x,y) = 3((y - 1) / 2)^2 + 6y^2. Simplifying this equation, we obtain G(x,y) = (3y^2 + 4y + 3) / 2.

To evaluate the integrals, we will use the parametric equations x = (y - 1) / 2 and y = 2x + 1.

Now, let's calculate each integral:

1. ∫C G(x,y)dx:
2. ∫C G(x,y)dy:
3. ∫C G(x,y)ds: