SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Calculus, 2ADV EQ-Bank 22

The diagram shows the graph of  \(y=\log _e(x+1)\)
 

  1. Express \(x\) as a function of \(y\).   (1 mark)

  2. Hence, or otherwise, find the exact area of the shaded region bounded by the curve  \(y=\log _e(x+1)\), the \(x\)-axis, and the line  \(x=3\).   (3 marks)

Show Answers Only

a.    \(x=e^y-1\)

b.    \(A=4 \ln 4-3 \ \text{u}^2\)

Show Worked Solution

a.    \(y=\ln (x+1) \ \Rightarrow \ x+1=e^y \ \Rightarrow \ x=e^y-1\)
 

b.    \(\text{Area of rectangle}=\ln 4 \times 3=3 \ln 4\)

\(\text{Find the area between curve and \(y\)-axis from  \(\ y=0\ \)  to  \(\ y=\ln 4\):}\)

\(A\) \(=\displaystyle \int_0^{\ln 4} e^y-1\, d y\)
  \(=\Big[e^y-y\Big]_0^{\ln 4}\)
  \(=\left(e^{\ln 4}-\ln 4\right)-(1)\)
  \(=4-\ln 4-1\)
  \(=3-\ln 4\)

 

\(\text{Shaded Area}\) \(=3 \ln 4-(3-\ln 4)\)
  \(=4 \ln 4-3 \ \text{u}^2\)