How to integrate the product of two functions

问题

Suppose I am seeking to integrate the following function from 0 to 10:

How would I accomplish this in R?

Functions

# Functional form
fn <- function(t) -100*(t)^2 + 20000

# First derivative w.r.t. t
fn_dt <- function(t) -200*t

# Density funciton phi
phi <- approxfun(density(rnorm(35, 15, 7)))

# Delta t
delta <- 5

回答1:

How about the following:

1. First off, we choose a fixed seed for reproducibility.

   # Density funciton phi
   set.seed(2017);
   phi <- approxfun(density(rnorm(35, 15, 7)))
   

2. We define the integrand.

   integrand <- function(x) {
       f1 <- -500 * x^2 + 100000;
       f2 <- phi(x);
       f2[is.na(f2)] <- 0;
       return(f1 * f2)
   }
   

   By default, approxfun returns NA if x falls outside the interval [min(x), max(x)]; since phi is based on the density of a normal distribution, we can replace NAs with 0.

3. Let's plot the integrand

   library(ggplot2);
   ggplot(data.frame(x = 0), aes(x)) + stat_function(fun = integrand) + xlim(-50, 50);
   

4. We use integrate to calculate the integral; here I assume you are interested in the interval [-Inf, +Inf].

   integrate(integrand, lower = -Inf, upper = Inf)
   #-39323.06 with absolute error < 4.6
   

来源：https://stackoverflow.com/questions/50828266/how-to-integrate-the-product-of-two-functions