punto 1,6,10.docx - 1 Hallar el rea que en el primer...

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1. Hallar el área que, en el primer cuadrante, está limitada por el eje X y por la siguiente función: 3 2 6 x x x y Sugerencia: Elabore la gráfica para una mejor comprensión del ejercicio. Está limitada en el eje X (0,3), entonces la integral nos queda 0 3 ( 6 x + x 2 x 3 ) Extraemos la constante 6 0 3 x + x 2 x 3 Aplicando la ley de potencias, nos queda Área a determinar
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6 0 3 xdx + 0 3 x 2 dx 0 3 x 3 dx 6 0 3 x 2 2 + x 3 3 x 4 4 Integrando nos queda ( 6 ) x 2 2 + x 3 3 x 4 4 ] 3 0 Evaluamos, pero debemos aplicar el teorema base a b f ( x ) dx = F ( b ) F ( a ) Reemplazando ( 3 ( 3 ) 2 + 3 3 3 3 4 4 ) ( 3 ( 0 ) 2 + 0 3 3 0 4 4 ) ¿ ( 27 + 9 20.25 ) ( 0 ) ¿ 15.75 esel áreaenel primer cuadrante 6. Hallar el volumen del sólido generado al rotar sobre el eje 1 x la región encerrada por la parábola 2 y x y la recta y x 2 (ver figura)
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v = π R 2 h π r 2 h v = πh R 2 r 2 R ¿ ( ¿ 2 r 2 ¿ ) dx ¿ dv = π a b ¿ ( y 2 ) −( 2 y ) 2 ¿ ¿ v = π 0 4 ¿ y 4 dy 4 y 2
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