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.

pg. 339 7.1.4

Find the area of the shaded region in the given graph:

To find the area between two curves on an interval from \(a\) to \(b\), find the area beneath the first curve \(f(x)\) and subtract from it the area beneath the second curve \(g(x)\) so that \(\int_{a}^{b} (f(x)-g(x))dx\).

ar.bet.cur = function(equation1, equation2, fromx, tox){
  
  # Plot max and mins
  max1 = optimize(equation1,c(fromx, tox),maximum=T)$objective
  max2 = optimize(equation2,c(fromx, tox),maximum=T)$objective
  overall.max = max(max1,max2)+1
  
  min1 = optimize(equation1,c(fromx, tox),maximum=F)$objective
  min2 = optimize(equation2,c(fromx, tox),maximum=F)$objective
  overall.min = min(min1,min2)-1
  
  # Plots
  curve(equation1, from=fromx, to=tox, n=300, ylim = c(overall.min, overall.max), xlab="X", ylab="Y", col="blue",lwd=2)
  curve(equation2, from=fromx, to=tox, n=300, ylim = c(overall.min, overall.max), add=TRUE, xlab="X", ylab="Y", col="red",lwd=2)
  
  legend( "topleft", inset = c(0,0), cex = .5,bty = "n", legend = c(equation1, equation2), text.col = c("red", "blue"), col = c("red", "blue"), pt.bg = c("red","blue"), pch = c(0,0))
  x = integrate(equation1, lower = fromx, upper = tox)
  matches = regmatches(x, gregexpr("[[:digit:]]+\\.*[[:digit:]]*", x))
  sol1 = as.numeric(matches$value)
  
  y = integrate(equation2, lower = fromx, upper = tox)
  matches2 = regmatches(y, gregexpr("[[:digit:]]+\\.*[[:digit:]]*", y))
  sol2 = as.numeric(matches2$value)
  
  sol = sol1-sol2
  sol = cat("Area between curves: ", sol)
  return(sol)
}

For problem 4, we have \(f(x)= \frac{1}{2}x+3\) and \(g(x)=\frac{1}{2}cos(x)+1\) from \(0\) to \(2\pi\).

equation1 = function(x){(1/2)*x+3}
equation2 = function(x){(1/2)*cos(x)+1}

ar.bet.cur(equation1, equation2, 0, 2*pi)

## Area between curves:  22.43598
## NULL

For problem 5, we have \(f(x)= -3x^3+3x+2\) and \(g(x)=x^2+x-1\) from \(-1\) to \(1\).

equation1 = function(x){-3*(x^3)+3*x+2}
equation2 = function(x){(x^2)+x-1}

ar.bet.cur(equation1, equation2, -1, 1)

## Area between curves:  2.666667
## NULL