AsignacionAreas

Asignacion 1

Estimar el area de una elipse cuyo semieje mayor a = 2 y semieje menor b = 1

Area de una elpise \(\pi*a*b\)

n = 100000
x = runif(n,0,2)
y = runif(n,0,1)

Ecuacion de una elipse

\[ \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 \]

d = ((((x^2)/4)+(y^2))<=1)
plot(x, y, pch = 16, asp = TRUE, col=d+1, cex=0.5)

100*table(d)/n

## d
##  FALSE   TRUE 
## 21.444 78.556

A_real = (pi*2)/4
A_est = (sum(d)/n) * 2
A_real

## [1] 1.570796

A_est

## [1] 1.57112

Asignacion 2

Hallar el area encerrada entre las curvas siguientes y compararla con la real calculada con integrales

\[ A = \int_{0}^{1}\sqrt{x}-x^{2} dx \] \[ A = \int_{0}^{1}{\sqrt{x}} dx -\int_{0}^{1}x^{2} dx \]

\[ A = \int_{0}^{1}{\sqrt{x}} dx -\int_{0}^{1}x^{2} dx \] \[A = \frac{2x^{3/2}}{3}|_0^{1}-\frac{x^3}{3}|_0^1\] \[A = \frac{2(1)^{3/2}}{3}-\frac{1^3}{3}\] \[ A = \frac{2}{3}-\frac{1}{3}\]

\[A = \frac{1}{3} = 0.333\]

n=100000
set.seed(123)
x=runif(n,0,1)
y=runif(n,0,1)

f=  (y<=sqrt(x) & y>=x^2)
plot(x,y,pch=20, cex = 0.5, col=f )
text(0.25,0.6,expression(sqrt(x)))
text(0.6,0.25,expression(x^2))

100*table(f)/n

## f
##  FALSE   TRUE 
## 66.874 33.126

a <- table(f)/n