Exercício de área entre curvas:
Determine a área limitada entre as curvas y=4x e y=x³+3x²
Resposta: 131/4
Respostas
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Primeiro temos que encontrar os pontos na qual as funções se interceptam:
![\left \{ {{y=4x} \atop {y=x^3+3x^2}} \right. \\ \\
4x = x^3 + 3x^2 \\ \\
x^3 + 3x^2 - 4x = 0 \\ \\
x(x^2 + 3x - 4) = 0 \to x' = 0 \\ \\
x^2 + 3x - 4 = 0 \\ \\
x'' = 1 \\
x''' = -4 \left \{ {{y=4x} \atop {y=x^3+3x^2}} \right. \\ \\
4x = x^3 + 3x^2 \\ \\
x^3 + 3x^2 - 4x = 0 \\ \\
x(x^2 + 3x - 4) = 0 \to x' = 0 \\ \\
x^2 + 3x - 4 = 0 \\ \\
x'' = 1 \\
x''' = -4](https://tex.z-dn.net/?f=+%5Cleft+%5C%7B+%7B%7By%3D4x%7D+%5Catop+%7By%3Dx%5E3%2B3x%5E2%7D%7D+%5Cright.++%5C%5C++%5C%5C+%0A4x+%3D+x%5E3+%2B+3x%5E2+%5C%5C++%5C%5C+%0Ax%5E3+%2B+3x%5E2+-+4x+%3D+0+%5C%5C++%5C%5C++%0Ax%28x%5E2+%2B+3x+-+4%29+%3D+0+%5Cto+x%27+%3D+0+%5C%5C++%5C%5C++%0Ax%5E2+%2B+3x+-+4+%3D+0+%5C%5C++%5C%5C+%0Ax%27%27+%3D+1+%5C%5C++%0Ax%27%27%27+%3D+-4)
Logo aplicando o teorema fundamental do cálculo:
![\left (\int\limits^1_0 {4x} \, dx - \int\limits^1_0 {x^3 + 3x^2} \, dx \right) + \left (\int\limits^0_{-4} {x^3 + 3x^2} \, dx - \int\limits^0_{-4} {4x} \, dx \right) \\ \\ \left (\int\limits^1_0 {4x} \, dx - \int\limits^1_0 {x^3 + 3x^2} \, dx \right) + \left (\int\limits^0_{-4} {x^3 + 3x^2} \, dx - \int\limits^0_{-4} {4x} \, dx \right) \\ \\](https://tex.z-dn.net/?f=+%5Cleft+%28%5Cint%5Climits%5E1_0+%7B4x%7D+%5C%2C+dx++-++%5Cint%5Climits%5E1_0+%7Bx%5E3+%2B+3x%5E2%7D+%5C%2C+dx+%5Cright%29+%2B++%5Cleft+%28%5Cint%5Climits%5E0_%7B-4%7D+%7Bx%5E3+%2B+3x%5E2%7D+%5C%2C+dx++-++%5Cint%5Climits%5E0_%7B-4%7D+%7B4x%7D+%5C%2C+dx+%5Cright%29+%5C%5C++%5C%5C+)
Encontrando as integrais indefinidas:
![\int {4x} \, dx = 4\int {x} \, dx = 4* \frac{x^2}{2} = 2x^2 \\ \\
\int {x^3 + 3x^2} \, dx = \int {x^3} \, dx + \int {3x^2} \, dx = \frac{x^4}{4} + 3* \frac{x^3}{3} = \frac{x^4}{4} + x^3 \int {4x} \, dx = 4\int {x} \, dx = 4* \frac{x^2}{2} = 2x^2 \\ \\
\int {x^3 + 3x^2} \, dx = \int {x^3} \, dx + \int {3x^2} \, dx = \frac{x^4}{4} + 3* \frac{x^3}{3} = \frac{x^4}{4} + x^3](https://tex.z-dn.net/?f=+%5Cint+%7B4x%7D+%5C%2C+dx++%3D++4%5Cint+%7Bx%7D+%5C%2C+dx++%3D++4%2A+%5Cfrac%7Bx%5E2%7D%7B2%7D++%3D+2x%5E2+%5C%5C++%5C%5C+%0A+%5Cint+%7Bx%5E3+%2B+3x%5E2%7D+%5C%2C+dx++%3D++%5Cint+%7Bx%5E3%7D+%5C%2C+dx++%2B++%5Cint+%7B3x%5E2%7D+%5C%2C+dx++%3D++%5Cfrac%7Bx%5E4%7D%7B4%7D+%2B+3%2A+%5Cfrac%7Bx%5E3%7D%7B3%7D++%3D++%5Cfrac%7Bx%5E4%7D%7B4%7D++%2B+x%5E3)
Logo temos:
![A =\left [ \left (2x^2 \right )\limits^1_0 - \left ( \frac{x^4}{4} + x^3 \right )\limits^1_0 \right ] + \left [ \left ( \frac{x^4}{4} + x^3 \right )\limits^0_{-4} - \left (2x^2 \right )\limits^0_{-4} \right ] A =\left [ \left (2x^2 \right )\limits^1_0 - \left ( \frac{x^4}{4} + x^3 \right )\limits^1_0 \right ] + \left [ \left ( \frac{x^4}{4} + x^3 \right )\limits^0_{-4} - \left (2x^2 \right )\limits^0_{-4} \right ]](https://tex.z-dn.net/?f=A+%3D%5Cleft+%5B+%5Cleft+%282x%5E2+%5Cright+%29%5Climits%5E1_0++-+%5Cleft+%28+%5Cfrac%7Bx%5E4%7D%7B4%7D++%2B+x%5E3+%5Cright+%29%5Climits%5E1_0+%5Cright+%5D+%2B+%5Cleft+%5B+%5Cleft+%28+%5Cfrac%7Bx%5E4%7D%7B4%7D+%2B+x%5E3+%5Cright+%29%5Climits%5E0_%7B-4%7D++-+%5Cleft+%282x%5E2+%5Cright+%29%5Climits%5E0_%7B-4%7D+%5Cright+%5D+)
Aplicando os limites (irei aplicar individualmente para ter espaço):
![\left [ \left (2x^2 \right )\limits^1_0 - \left ( \frac{x^4}{4} + x^3 \right )\limits^1_0 \right ] \\ \\
\ [(2*1^2 - 2*0^2) ] - \leftt [( \frac{1^4}{4} + 1^3 ) - ( \frac{0^4}{4} + 0^4)] \\ \\
2 - ( \frac{1}{4} + 1) = 2 - \frac{5}{4} = \boxed{ \frac{3}{4} } \left [ \left (2x^2 \right )\limits^1_0 - \left ( \frac{x^4}{4} + x^3 \right )\limits^1_0 \right ] \\ \\
\ [(2*1^2 - 2*0^2) ] - \leftt [( \frac{1^4}{4} + 1^3 ) - ( \frac{0^4}{4} + 0^4)] \\ \\
2 - ( \frac{1}{4} + 1) = 2 - \frac{5}{4} = \boxed{ \frac{3}{4} }](https://tex.z-dn.net/?f=%5Cleft+%5B+%5Cleft+%282x%5E2+%5Cright+%29%5Climits%5E1_0++-+%5Cleft+%28+%5Cfrac%7Bx%5E4%7D%7B4%7D++%2B+x%5E3+%5Cright+%29%5Climits%5E1_0+%5Cright+%5D+%5C%5C++%5C%5C++%0A%5C+%5B%282%2A1%5E2+-+2%2A0%5E2%29+%5D+-+%5Cleftt+%5B%28+%5Cfrac%7B1%5E4%7D%7B4%7D+%2B+1%5E3+%29+-+%28+%5Cfrac%7B0%5E4%7D%7B4%7D++%2B+0%5E4%29%5D+%5C%5C++%5C%5C++%0A2++-+%28+%5Cfrac%7B1%7D%7B4%7D++%2B+1%29+%3D+2+-++%5Cfrac%7B5%7D%7B4%7D+%3D+%5Cboxed%7B+%5Cfrac%7B3%7D%7B4%7D+%7D)
![\left [ \left ( \frac{x^4}{4} + x^3 \right )\limits^0_{-4} - \left (2x^2 \right )\limits^0_{-4} \right ] \\ \\ \\ \ [( \frac{0^4}{4} - 0^3 ) - (\frac{(-4)^4}{4} + (-4)^3)] - [(2*0^2 - 2*(-4)^2) \\ \\ -(64 - 64) -(-32) = \boxed{32} \left [ \left ( \frac{x^4}{4} + x^3 \right )\limits^0_{-4} - \left (2x^2 \right )\limits^0_{-4} \right ] \\ \\ \\ \ [( \frac{0^4}{4} - 0^3 ) - (\frac{(-4)^4}{4} + (-4)^3)] - [(2*0^2 - 2*(-4)^2) \\ \\ -(64 - 64) -(-32) = \boxed{32}](https://tex.z-dn.net/?f=+%5Cleft+%5B+%5Cleft+%28+%5Cfrac%7Bx%5E4%7D%7B4%7D+%2B+x%5E3+%5Cright+%29%5Climits%5E0_%7B-4%7D+-+%5Cleft+%282x%5E2+%5Cright+%29%5Climits%5E0_%7B-4%7D+%5Cright+%5D+%5C%5C+%5C%5C+%5C%5C+%5C+%5B%28+%5Cfrac%7B0%5E4%7D%7B4%7D+-+0%5E3+%29+-+%28%5Cfrac%7B%28-4%29%5E4%7D%7B4%7D+%2B+%28-4%29%5E3%29%5D+-+%5B%282%2A0%5E2+-+2%2A%28-4%29%5E2%29+%5C%5C+%5C%5C+-%2864+-+64%29+-%28-32%29+%3D++%5Cboxed%7B32%7D)
Assim temos a área limitada pelas funções:
Logo aplicando o teorema fundamental do cálculo:
Encontrando as integrais indefinidas:
Logo temos:
Aplicando os limites (irei aplicar individualmente para ter espaço):
Assim temos a área limitada pelas funções:
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