1
$\begingroup$

Find the area between the curves $y=\ln(x), y=1, y=-1, y^2 = x+2$

Doing a sketch on Desmos, I see this graph

enter image description here

The beige coloring is the area in between the graphs

It is obvious I have to break up this graph and calculate their respective areas seperately, and then add. How do I know where to break up though?

enter image description here

This is where I have decided to break up the graphs and calculate their area and add it all up. Is this a reasonable breaking structure, or am I doing too much work?

  • 1
    Use integration about $y$ – 2017-01-29
  • 0
    Would it just be one integral? – 2017-01-29
  • 0
    Would $\int _{-1}^1\left(e^y-\left(y^2-2\right)\right)dy$ be the integral? – 2017-01-29
  • 0
    Find answer.... – 2017-01-29
  • 0
    I found it to be about $5.7$ but I have no way of checking if I'm right – 2017-01-29
  • 0
    Correct. I checked $5.68374$ – 2017-01-29
  • 0
    Alright, cool. If you would like to put your answer in the comments I can mark as correct – 2017-01-29

1 Answers 1

4

What you want to do is integrate with respect to $y$. Turn the graph in your head if you need to. What you get is $$\int _{-1}^1\left(e^y-\left(y^2-2\right)\right)dy$$
As you noted yourself in the comments. To evaluate this integral we can split into three separate integrals; the first is trivial because $\frac{d}{dx} e^x = e^x$, and the second/third require the power rule of integration, i.e. $\int x^n = \frac{x^{n+1}}{n+1}$

  • 0
    If I were to integrate with respect with $x$, how many integrals would I need? – 2017-01-29
  • 0
    Hmm.. For the Cyan and green part, integrate $y=1$ from $-1$ to $e$ and double the value of that integral. Then subtract the integral of $\ln(x)$ from $\frac{1}{e}$ to $e$. – 2017-01-29
  • 0
    @KSplitX For the gray/yellow area, integrate $y=\sqrt{x+2}$ from $-2$ to $-1$ and double that integral. So in total you *can* solve this with three integrals if you integrate with respect to $x$, it is just a bit harder – 2017-01-29
  • 0
    @KSplitX You could also solve this using Green's Theorem, which could be seen as a single (line) integral, or you could solve using a double integral. So anywhere from $1$ to $3$ integrals ought to be used, depending on how complicated you want to go – 2017-01-29