ATPS De Cálculo III

ATPS de Cálculo III

Etapa 1: Integral Definida, Integral Indefinida

Passo 1:

Exemplos de Integral definida:

∫_1^3▒〖(x^2+〗 4)□(24&dx)→x^3/3+4x=

(〖(3)〗^3/3+4*3)-(〖(1)〗^3/3+4*1)=

(9+12)-(1/3+4)=21-((1+12)/3)=

21-13/3=(63-13)/3=▭(50/3) ∫_2^4▒〖x^2+〗 2x□(24&dx)→x^3/3+(2x^2)/2=

(〖(4)〗^3/3+〖2*(4)〗^2/2)-(〖(2)〗^3/3+〖2*(2)〗^2/2)=

(64/3+16)-(8/3+4)=((64-48)/3)-((8+12)/3)=

16/3-20/3=(16-20)/3=▭(-4/3)

∫_0^1▒〖(x^3-6x-8〗)□(24&dx)→x^4/4-(6x^2)/2+8x=

(〖(1)〗^4/4-(6*(1)^2)/2+8*(1))-

(((〖0)〗^4)/4-(6〖*(0)〗^2)/2+8*(0))=

(1/4-6/2+8)-0=(1-12+32)/4=▭2 ∫_1^4▒dx/√x→2√x=

(2√4)-(2√1)=4-2=▭2

Exemplos de Integral indefinida:

∫▒〖4x^3 □(24&dx)〗→

∫▒〖〖4x〗^4/4=〗 ▭(x^4+C) ∫▒〖3e^x+1/4x-sen⁡x □(24&dx)〗→

3e^x+1/4 ln⁡|x|-(-cos⁡〖x)+C〗

▭(3e^x+1/4 ln⁡|x|+cos⁡〖x+C〗 )

∫▒〖5x^2+7x+2□(24&dx)〗→

▭((5x^3)/3+(7x^2)/2+2x+C) ∫▒〖cos⁡x-sen⁡x □(24&dx)〗→

sen⁡x-(-cos⁡〖x)+C〗

▭(sen⁡x+cos⁡〖x+C〗 )

Exemplos de Cálculo de área:

Determine a área:

A=∫_0^5▒〖5x+x^2 〗 □(24&dx)→A=(5x^2)/2-x^3/3=

A=((5*〖(5)〗^2)/2-〖(5)〗^3/3)-((5*〖(0)〗^2)/2-〖(0)〗^3/3)=

A=(5^3/2-5^3/3)-0→A=(15-10)/6=▭(A=5/6 u.a)

Dada a função y=x calcular a área sob o gráfico de x=0 e x=-3:

A=∫_0^3▒x □(24&dx)→A=x^2/2=

A=(〖(3)〗^2/2)-((0)^2/2)→▭(A=9/2 u.a)

Calcular a área da região limitada inferiormente pela curva y=x^2-3x+2:

A=∫_1^2▒x^2 -3x+2□(24&dx)→A=x^3/3-(3x^2)/2+2x=

A=(((〖2)〗^3)/3-(3〖*(2)〗^2)/2+2*(2))-(((〖1)〗^3)/3-(3〖*(1)〗^2)/2+2*(1))=

A=(8/3-6+4)-(1/3-3/2+2)→A=((8-18+12)/3)-((2-9+12)/6)=

A=(2/3)-(5/6)→A=(4-5)/6=▭(A=1/6 u.a.)

Determine a área:

A=∫_1^3▒x^2 □(24&dx)→A=x^3/3=

A=(〖(3)〗^3/3)-((1)^3/3)→A=9-1/3=

A=(27-1)/3→▭(A=26/3 u.a.)