R Notebook

# =============================================
# Exercise A - Normal Distribution
# Simpson's 1/3 Rule with n = 4
# =============================================

# Fungsi PDF Standard Normal
f <- function(x) {
  1 / sqrt(2 * pi) * exp(-x^2 / 2)
}

# Parameter
a <- 0
b <- 1
n <- 4 # harus genap
h <- (b - a) / n

# Titik-titik x
x <- seq(a, b, length.out = n + 1)

# Hitung f(x) di setiap titik
y <- f(x)

# Simpson's 1/3 Rule
simpson <- (h / 3) * (y[1] + 4 * sum(y[seq(2, n, by = 2)]) +  2 * sum(y[seq(3, n-1, by = 2)]) + y[n+1])

# Hasil
cat("=== HASIL SIMPSON'S 1/3 RULE (n=4) ===\n")

=== HASIL SIMPSON'S 1/3 RULE (n=4) ===

cat("Approksimasi P(0 ≤ X ≤ 1) =", simpson, "\n\n")

Approksimasi P(0 ≤ X ≤ 1) = 0.3413555 

# Nilai eksak menggunakan pnorm (untuk perbandingan)
exact <- pnorm(1) - pnorm(0)
cat("Nilai eksak (pnorm(1) - pnorm(0)) =", exact, "\n")

Nilai eksak (pnorm(1) - pnorm(0)) = 0.3413447 

cat("Error absolut =", abs(simpson - exact), "\n")

Error absolut = 1.074179e-05 

cat("Error relatif (%) =", abs(simpson - exact)/exact * 100, "%\n")

Error relatif (%) = 0.003146903 %

# =============================================
# Exercise B - Exponential Distribution
# Trapezoidal Rule & Simpson's 1/3 Rule (n=10)
# =============================================

# Fungsi PDF Exponential (λ=1)
f <- function(x) {
  exp(-x)
}

# Parameter
a <- 0
b <- 2
n <- 10 # harus genap
h <- (b - a) / n

# Titik-titik x dan nilai f(x)
x <- seq(a, b, length.out = n + 1)
y <- f(x)

# ======================
# 1. Trapezoidal Rule
# ======================
trapezoidal <- (h/2) * (y[1] + 2 * sum(y[2:n]) + y[n+1])

# ======================
# 2. Simpson's 1/3 Rule
# ======================
simpson <- (h/3) * (y[1] + 4 * sum(y[seq(2, n, by = 2)]) + 2 * sum(y[seq(3, n-1, by = 2)]) + y[n+1])

# ======================
# Hasil dan Perbandingan
# ======================
exact <- 1 - exp(-2)   # nilai eksak dari integral

cat("=== HASIL PERHITUNGAN n = 10 ===\n")

=== HASIL PERHITUNGAN n = 10 ===

cat("Trapezoidal Rule :", trapezoidal, "\n")

Trapezoidal Rule : 0.867545 

cat("Simpson's 1/3 Rule :", simpson, "\n")

Simpson's 1/3 Rule : 0.8646724 

cat("Nilai Eksak (1 - e^{-2}) :", exact, "\n\n")

Nilai Eksak (1 - e^{-2}) : 0.8646647 

cat("Error Trapezoidal :", abs(trapezoidal - exact), "\n")

Error Trapezoidal : 0.002880296 

cat("Error Simpson :", abs(simpson - exact), "\n")

Error Simpson : 7.649462e-06 

cat("Error relatif Trapezoidal (%):", abs(trapezoidal - exact)/exact * 100, "%\n")

Error relatif Trapezoidal (%): 0.3331113 %

cat("Error relatif Simpson (%) :", abs(simpson - exact)/exact * 100, "%\n")

Error relatif Simpson (%) : 0.0008846738 %

```