Discussion Week 14
Bonnie Cooper
7.1.18
Find the total area enclosed by the functions \(f\) and \(g\) where:
\[f(x) = -x^3 +5x^2+2x+1\] \[g(x) = 3x^2+ x+3\]
to start, let’s visualize the functions:
library( ggplot2 )
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
fun.1 <- function(x) -x^3 +5*x^2 + 2*x + 1
fun.2 <- function(x) 3*x^2 + x +3
p + stat_function(fun = fun.1) + stat_function(fun = fun.2) + xlim(-2,3)
The two functions appear to enclose a bounded regions between the x-values of -1,1 &2 (ish). Now to solve more definitively where the two functions intersect (have the same values). \[-x^3 +5x^2+2x+1 = 3x^2+ x+3\rightarrow\] use uniroot to find the x value of the intersections for each interval that approximates where a crossing occurs based on the visualization:
root1 <- uniroot(function(x) fun.1(x) - fun.2(x) , c(-1.5,-0.5), tol=1e-8)
root2 <- uniroot(function(x) fun.1(x) - fun.2(x) , c(0.5,1.5), tol=1e-8)
root3 <- uniroot(function(x) fun.1(x) - fun.2(x) , c(1.5,2.5), tol=1e-8)
paste( 'Three roots were approximated to have the x-values:', root1$root, ',', round( root2$root,6),',' , round(root3$root,6))## [1] "Three roots were approximated to have the x-values: -1 , 1 , 2"solving for the y-values:
yn1 <- fun.1( -1 )
y1 <- fun.1( 1 )
y2 <- fun.1( 2 )
paste( 'The y-values for the roots:', yn1, ',', y1, ',', y2 )## [1] "The y-values for the roots: 5 , 7 , 17"Therefore, find the area of the regions bounded by the points ( -1,5 ), ( 1,7 ) & ( 2,17 ).
To do this, we will use Theorem 7.1.1 (pg354) to find the difference of the ingrated functions with respect to x from values for both bounded regions from \(x=-1\) to \(x=1\), and from \(x=1\) to \(x=2\)
\[\mbox{Theorem 7.1.1:}\int_a^b(f(x)-g(x))dx\]
Area of bounded region from \(x=-1\) to \(x=1\) \[\int_{-1}^{1}((3x^2 + x + 3) - (-x^3 + 5x^2 +2x+ 1))dx \rightarrow\] \[\int_{-1}^{1}(3x^3-2x^2 -x+2)dx \rightarrow\] \[\frac{3x^4}{4}-\frac{2x^3}{3} - \frac{x^2}{2} + 2x |_{-1}^1 \] Now to take the difference of the expression evaluated at -1, 1:
fun.3 <- function(x) (3*x^4)/4 -(2*x^3)/3 - (x^2)/2 + 2*x
arean1t1 <- fun.3( 1 ) - fun.3( -1 )
paste( 'Area bounded by curves =', arean1t1 )## [1] "Area bounded by curves = 2.66666666666667"Area of bounded region from \(x=1\) to \(x=2\) \[\int_{1}^{2}((-x^3 + 5x^2 +2x+ 1)-(3x^2 + x + 3))dx \rightarrow\] \[\int_{1}^{2} (-x^3 +2x^2+x-2)dx \rightarrow\] \[-\frac{x^4}{4} + \frac{2x^3}{3} + \frac{x^2}{2} - 2x |_{1}^2\]
Now to take the difference of the expression evaluated at 1, 2:
fun.4 <- function(x) (-1*x^4)/4 +(2*x^3)/3 + (x^2)/2 -2*x
area1t2 <- fun.4( 2 ) - fun.4( 1 )
paste( 'Area bounded by curves =', area1t2 )## [1] "Area bounded by curves = 0.416666666666667"The total area bounded by the two functions is the sum of these two subregions:
## [1] "The total bounded area = 3.08333333333333"