Problem

Using R, provide the solution for any exercise in either Chapter 4 or Chapter 7 of the calculus textbook. If you are unsure of your solution, post your concerns.

Ex. 7.1.14 (Text: Apex Calculus by Hartman)

Find the total area enclosed by the functions f and g.

\(f(x) = x^2 - 3x + 2, \space g(x) = -3x + 3\)

Ans:

https://www.mathsisfun.com/data/function-grapher.php#

Solving the equations for f(x) and g(x), we get x2 - 1 = 0, x = -1 or 1, which is confirmed from the graphical approach above.

f <- function (x) {x^2 - 3*x + 2}
g <- function (x) {-3*x + 3}
curve(f, xlim = c(-1,3), ylim = c(-1,7), ylab = 'f(x), g(x)', col='red', sub = paste0("f(x)= x^2-3x+2", "   g(x)= -3x+3"))
curve(g, col='blue', add = TRUE)

The function to integrate should be g(x) - f(x) to get the total area enclosed, i.e., we’ve to solve for

\(\int_{-1}^{1} -x^2+1dx\)

integrand <- function(x){-x^2 + 1}
area <- stats::integrate(integrand, -1, 1)
area
## 1.333333 with absolute error < 1.5e-14

The answer is 4/3.