# Area between two functions?

Find the area between the functions $x+y = 2$ and $x + 4 = y^2$.

The question is relatively simple:

The area between the functions is:

$$\int^{2}_{-3} 2-y-y^2+4 \text{ }dy$$

But can the above area be found by integrating a function with respect to $dx$? As in can the area between the functions be defined as the integration of functions of $x$? (i.e. integration of $f(x)$ and $g(x))$?

• This depends how you express the functions: either $y=2-x$ and $y=|\sqrt{4+x}|$ or $x=2-y$ and $x=y^2-4$ and then you have to change the bounds of integration accordingly – Alex Jan 23 '14 at 1:51
• But the second function isn't $y = |\sqrt{5+x}|$, its $y^2 = \sqrt{5+x}$ – dfg Jan 23 '14 at 2:06
• @Alex The functions are different because the first one is half a hyperbola, the second is a full one – dfg Jan 23 '14 at 2:06
• I think you mean a parabola... – colormegone Jan 23 '14 at 2:13
• @dfg: if $y^2=x$ then when you take the square root you get $y=\pm \sqrt{x} = |\sqrt{x}|$ – Alex Jan 23 '14 at 2:17

You would need to break the calculation into two separate integrals, one between the "upper" and "lower" arms of the parabola over the interval $\ [ -4, 0] \$ , the other between the line and the "lower" arm on the interval $\ [ 0, 5 ] \ .$ So the area would be found from
$$\int_{-4}^0 \ [ \ \sqrt{x+4} \ ] \ - \ [ \ -\sqrt{x+4} \ ] \ \ dx \ \ + \ \ \int_0^5 \ [ \ 2 - x \ ] \ - \ [ \ -\sqrt{x+4} \ ] \ \ dx \ .$$