Calcule ∫ C F · dr onde F (x, y, z) = (y^2z + 2xz^2)i + (2xyz)j + (xy^2 + 2x^2z)k e C : x = √t, y = t+ 1, z = t^2, 0 ≤ t ≤ 1.

Para calcular a integral ∫ C F · dr, onde F(x, y, z) = (y^2z + 2xz^2)i + (2xyz)j + (xy^2 + 2x^2z)k e C é a curva definida por x = √t, y = t+1, z = t^2, com 0 ≤ t ≤ 1, podemos seguir os seguintes passos: 1. Calcule as derivadas de x, y e z em relação a t: dx/dt = 1/(2√t) dy/dt = 1 dz/dt = 2t 2. Substitua as derivadas e as componentes de F na integral: ∫ C F · dr = ∫[0,1] [(y^2z + 2xz^2)(dx/dt) + (2xyz)(dy/dt) + (xy^2 + 2x^2z)(dz/dt)] dt 3. Substitua as componentes de x, y e z na integral: ∫ C F · dr = ∫[0,1] [((t+1)^2(t^2) + 2(√t)(t^2)^2)(1/(2√t)) + (2(√t)(t+1)(t^2)) + ((√t)(t+1)^2 + 2(√t)^2(t^2))(2t)] dt 4. Simplifique e resolva a integral: ∫ C F · dr = ∫[0,1] [(t^4 + 2t^4)(1/(2√t)) + (2t√t(t+1)(t^2)) + (√t(t+1)^2 + 2t^2)(2t)] dt ∫ C F · dr = ∫[0,1] [(3t^4)/(2√t) + 2t^2√t(t+1)(t^2) + 2t^3(t+1)^2 + 4t^3] dt 5. Integre termo a termo: ∫ C F · dr = [(3t^5)/(10√t) + (2t^5√t(t+1))/(3/2) + (2t^4(t+1)^2)/(4) + (4t^4)/(4)] |[0,1] 6. Substitua os limites de integração: ∫ C F · dr = [(3(1)^5)/(10√1) + (2(1)^5√1(1+1))/(3/2) + (2(1)^4(1+1)^2)/(4) + (4(1)^4)/(4)] - [(3(0)^5)/(10√0) + (2(0)^5√0(0+1))/(3/2) + (2(0)^4(0+1)^2)/(4) + (4(0)^4)/(4)] 7. Simplifique e calcule os valores: ∫ C F · dr = [3/10 + 2/3 + 2 + 1] - [0 + 0 + 0 + 0] ∫ C F · dr = 3/10 + 2/3 + 2 + 1 ∫ C F · dr = 3/10 + 6/9 + 2 + 1 ∫ C F · dr = 27/90 + 60/90 + 180/90 + 90/90 ∫ C F · dr = 357/90 ∫ C F · dr = 3.9667 Portanto, o valor da integral ∫ C F · dr é aproximadamente 3.9667.