Diferenças

Aqui você vê as diferenças entre duas revisões dessa página.

--- exercicios:calc_int [2012/05/05 17:48] – [Integral definidas] adalardo
+++ exercicios:calc_int [2024/01/09 18:18] (atual) – edição externa 127.0.0.1
@@ Linha 12: / Linha 12: @@
 ^ Ache as derivadas e em seguida as antiderivadas ^^^^^^
-  - $$ f(x) = exp(x) + x^7$$
+  - $ f(x) = exp(x) + x^7$
-  - $$ f(x) = x + sin(x) $$
+  - $ f(x) = x + sin(x) $
-  - $$ f(x) = 5x^3 + 2$$
+  - $ f(x) = 5x^3 + 2$
-  - $$ f(x) = cos(x) + sin(x) $$
+  - $ f(x) = cos(x) + sin(x) $
-  - $$ f(x) = x^2 + x^3cos(x)$$
+  - $ f(x) = x^2 + x^3cos(x)$
-  - $$ f(x) = exp(x) ln(x) $$
+  - $ f(x) = exp(x) ln(x) $
-  - $$ f(x) = x^5sin(x)$$
+  - $ f(x) = x^5sin(x)$
-  - $$ f(x) = 1/x $$
+  - $ f(x) = \frac{1}{x} $
-  - $$ f(x) = 1/x^2 $$
+  - $ f(x) = \frac{1}{x^2} $
-  - $$ f(x) = exp(x)/x $$
+  - $ f(x) = \frac{exp(x)}{x} $
-  - $$ f(x) = sin(x)/x^2$$
+  - $ f(x) = \frac{sin(x)}{x^2}$
@@ Linha 38: / Linha 38: @@
-===== Integral definidas =====
+===== Integrais definidas =====
 {{:exercicios:area_x2.jpeg?250  |}}
 Podemos pensar a integral definida como a área resultante sob a curva da função em um dado intervalo.
-Vamos visualizar isso graficamente com a nossa já conhecida função quadratica $f(x)=x^2$ a área no intervalo de 0 até 1.
+Vamos visualizar isso graficamente com a nossa já conhecida função quadratica $f(x)=x^2$ a área no intervalo de 0 até 1. Que em notação matemática é representado como:
-$\int_0^1 f(x)~dx$
+$\int_0^1 f(x)~dx$
-FIXME
+==== Área Aproximada ====
+Vamos tentar resolver o problema de forma bastante intuitíva e com as ferramentas que temos ((Cuidado com o [[http://en.wikipedia.org/wiki/Law_of_the_instrument|Martelo de Maslow]]: "... se o único instrumento que tem é um martelo, todos o problemas parecem pregos!")). Não sabemos calcular a área sob curvas, apenas áreas de figuras geométricas regulares. Vamos então, transformar a curva em retângulos contíguos e calcular a somatória da área desses retângulos!
+Primeiro vamos desenhar o gráfico acima do nosso problema.
-==== Código ====
 <code>
 ##############################
@@ Linha 56: / Linha 57: @@
 ## no intervalo 0 a 1
 #############################
+par(mfrow=c(2,2))
 seq.x=seq(0,1.5, by=0.1)
 seq.y=seq.x^2
 plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= "Função x^2", xlab="x", ylab="y")
-#plot(seq.x,seq.y, type="l")
 abline(v=0, lty=2)
 abline(h=0, lty=2)
@@ Linha 66: / Linha 66: @@
 seq.y1=seq.x1^2
 polygon(c(1,0,seq.x1,1), c(0,0,seq.y1,0),col="red")
+title(sub=paste("Área= ??"))
 #savePlot("area_x2.jpeg", type="jpeg")
+</code>
+=== Cálculo da Área ===
+<code>
+#############################
+#### Aproximação da Área ###
+###########################
+n.seq1=length(seq.x1)
+plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= "Altura Mínima",xlab="x", ylab="y")
+abline(v=0, lty=2)
+abline(h=0, lty=2)
+abline(v=1, lty=2)
+barplot(height=seq.y1[-n.seq1],width=0.1, space=0, col="red", add=TRUE, yaxt="n")
+#################################
+## calculo da área dos retângulos
+##############################
+h1=seq.y1[-n.seq1]
+(ar1= sum(h1*0.1))
+title(sub=paste("Área=",ar1))
+</code>
+=== Outra Solução ===
+<code>
+################################
+## Altura da área a esquerda
 ###############################
-#### Calculo dos retângulos ###
+plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= "Altura Máxima", xlab="x", ylab="y")
+abline(v=0, lty=2)
+abline(h=0, lty=2)
+abline(v=1, lty=2)
+barplot(height=seq.y1[-1],width=0.1, space=0, col="red", add=TRUE,, yaxt="n")
+lines(seq.x,seq.y)
+#################################
+## calculo da área dos retângulos
+################################
+h2=seq.y1[-1]
+(ar2= sum(h2*0.1))
+title(sub=paste("Área=",ar2))
+</code>
+=== Altura Média ===
+<code>
+################################
+## Altura da área na media
 ###############################
-x11()
+plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= "Altura Média", xlab="x", ylab="y")
-n.seq=length(seq.x1)
-plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= "Função x^2", xlab="x", ylab="y")
 abline(v=0, lty=2)
 abline(h=0, lty=2)
 abline(v=1, lty=2)
-barplot(height=seq.y1[-n.seq],width=0.1, space=0, col="red", add=TRUE)
+barplot(height=diff(seq.y1)/2+seq.y1[-n.seq1],width=0.1, space=0, col="red", add=TRUE, yaxt="n")
+lines(seq.x,seq.y)
+#################################
+## calculo da área dos retângulos
+################################
+h3=diff(seq.y1)/2+seq.y1[-n.seq1]
+(ar3= sum(h3*0.1))
+title(sub=paste("Área=",ar3))
+################################
+</code>
+==== Diminuindo os intervalos ====
+Agora vamos diminuir os intervalos do eixo x, a partir da árean estimada para a altura média do retângulo no intervalo. Esse processo é o mesmo que dizer que o intervalo tende a zero $\Delta x \to 0$, em outras palavras estamos buscando a somatória de limites. Podemos formular dessa forma:
+$$\int_a^b f(x)~dx = \lim\limits_{\Delta x \to 0} \sum\limits_{i=1}^n f(x_i^*)\Delta x_i$$
+=== $d_x=0.1$ ===
+<code>
+##################################################
+## DIMINUNIDO O INTERVALO (BASE) DOS RETÂNGULOS ##
+##################################################
 x11()
-n.seq=length(seq.x1)
+par(mfrow=c(2,2))
-plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= "Função x^2", xlab="x", ylab="y")
+plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= "f(x)=x^2\t ; dx=0.1", xlab="x", ylab="y")
 abline(v=0, lty=2)
 abline(h=0, lty=2)
 abline(v=1, lty=2)
-barplot(height=seq.y1[-n.seq],width=0.1, space=0, col="red", add=TRUE)
+barplot(height=diff(seq.y1)/2+seq.y1[-n.seq1],width=0.1, space=0, col="red", add=TRUE, yaxt="n")
+lines(seq.x,seq.y)
+title(sub=paste("Área=",ar3))
 </code>
-===== Densidade =====
+=== $d_x=0.05$ ===
+<code>
+##############
+### dx=0.05 ##
+##############
+dx=0.05
+seq.05= seq(0,1, by=dx)
+seq.05y=seq.05^2
+plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= paste("dx=", dx), xlab="x", ylab="y")
+abline(v=0, lty=2)
+abline(h=0, lty=2)
+abline(v=1, lty=2)
+barplot(height=diff(seq.05y)/2+seq.05y[-length(seq.05y)],width=dx, space=0, col="red", add=TRUE, yaxt="n")
+lines(seq.x,seq.y)
+#################################
+## calculo da área dos retângulos
+################################
+h4=diff(seq.05y)/2+seq.05y[-length(seq.05y)]
+(ar4= sum(h4*dx))
+title(sub=paste("Área=",ar4))
+</code>
+=== $d_x=0.01$ ===
+<code>
+##############
+### dx=0.01 ##
+##############
+dx=0.01
+seq.01= seq(0,1, by=dx)
+seq.01y=seq.01^2
+plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= paste("dx=", dx), xlab="x", ylab="y")
+abline(v=0, lty=2)
+abline(h=0, lty=2)
+abline(v=1, lty=2)
+barplot(height=diff(seq.01y)/2+seq.01y[-length(seq.01y)],width=dx, space=0, col="red", add=TRUE, yaxt="n")
+lines(seq.x,seq.y)
+#################################
+## calculo da área dos retângulos
+################################
+h5=diff(seq.01y)/2+seq.01y[-length(seq.01y)]
+(ar5= sum(h5*dx))
+title(sub=paste("Área=",ar5))
+</code>
+=== $d_x=0.001$ ===
+<code>
+##############
+### dx=0.001 ##
+##############
+dx=0.001
+seq.001= seq(0,1, by=dx)
+seq.001y=seq.001^2
+plot(seq.x,seq.y, type="l", bty="l", cex.lab=1.5, cex.axis=1.2, main= paste("dx=", dx), xlab="x", ylab="y")
+abline(v=0, lty=2)
+abline(h=0, lty=2)
+abline(v=1, lty=2)
+barplot(height=diff(seq.001y)/2+seq.001y[-length(seq.001y)],width=dx, space=0, col="red", add=TRUE, yaxt="n")
+lines(seq.x,seq.y)
+#################################
+## calculo da área dos retângulos
+################################
+h6=diff(seq.001y)/2+seq.001y[-length(seq.001y)]
+(ar6= sum(h6*dx))
+title(sub=paste("Área=",ar6))
+</code>
+===== Maxima =====
+{{:exercicios:maximalogo.png?200  |}}
+Vamos integrar algumas equações no Maxima. Abra o arquivo {{:exercicios:integral.wxm|}} e aplique a integral nas funções apresentadas no roteiro.
+===== Exercicios =====
+Siga agora para a página de [[questionario:int|exercícios]] de integrais.