Cálculo de Área com Integrais Definidas: Exercícios Resolvidos

1 2 3 4 5 6 7 8 Desenhe o conjunto A dado e calcule a área A é o conjunto do plano limitado pelo eixo 0x pelo gráfico 𝑦 𝑥3 𝑥 1 𝑥 1 A é conjunto do plano limitado pela reta y 0 e pelo gráfico de 𝑦 𝑥3 𝑥 0 𝑥 2 A é o conjunto de todos xy tais que 𝑥 0 e 1 𝑥2 𝑦 5 4𝑥2 Yxx2 2x so λx2 2x 0 x0 x2 Ψ0 20 20 0 ½ 3 Ψ1 12 2 1 1 2 1 Ψx 2 0 minore Ψ appartient a R 2 et The without any little circle on top C it is a parabola and the graph is a convex function it has a minimum in the graph of the first case the concavity is positive the second case remains a minimum a 16 6 k Ψxλxx1x2 Gxxx1x2 Ψx 3x2 4x 1 Gxxx1x2 Ψx 3x2 4x 1 integral from 0 to 2 xx1x2 dx integral from 0 to 1 xx1x2 dx integral from 1 to 2 xx1x2 dx I1 I2 I1 integral from 0 to 1 xx1x2 dx integral from 0 to 1 x3 3x2 2x dx x44 x3 x2 evaluated from 0 to 1 ¼ 1 1 0 ¼ I2 integral from 1 to 2 xx1x2 dx I2 similar steps 94 calculation I1 I2 ¼ 94 84 2 b 0 λ 2 0 y x2 x A integral from 0 to 2 integral from 0 to x2 x dy dx A integral from 0 to 2 x2 x dx x33 x22 evaluated from 0 to 2 83 2 83 63 23 Volume of an area bounded by curves c image of a curve ψx φx fx gx fx sinx gx cosx φx y x φx y x ψx y 0 minimum A double integral dx dy over some limits calculated A complex steps not fully visible d Area calculations with integrals and steps expanding out with a final answer