Calculate the area of this pond with functions given for the perimeter

[tomwilliam]

Homework Statement

See image below. Trying to calculate area of a pond using the functions given for the upper and lower boundaries

Relevant Equations

The equation referred to in the booklet is the definite integral from a to b of f(x) wrt dx = F(b) - F(a)

So the solution is obviously given here, I'm just trying to understand it. I thought that integrating f(x) from -5 to 5 would give the area under the curve (including the areas below the "pond" at the edges of the image but above y=0. I don't really understand why we are subtracting the integral of g(x).
Any help much appreciated!
Thanks

 

[Hill]

Homework Statement: See image below. Trying to calculate area of a pond using the functions given for the upper and lower boundaries
Relevant Equations: The equation referred to in the booklet is the definite integral from a to b of f(x) wrt dx = F(b) - F(a)

View attachment 346662

So the solution is obviously given here, I'm just trying to understand it. I thought that integrating f(x) from -5 to 5 would give the area under the curve (including the areas below the "pond" at the edges of the image but above y=0. I don't really understand why we are subtracting the integral of g(x).
Any help much appreciated!
Thanks

To get the blue area, you need to subtract from the $\int_{-5}^5 f(x) dx$ the areas $a$ and $b$ and to add to it the area $c$:

This is what subtracting $\int_{-5}^5 g(x) dx$ does.

 

[jbriggs444]

Another way to understand the same result is to imagine the area of the pond as a bunch of [blue-shaded] vertical strips, all side by side.

The $y$ extent of the strip at $x$ is given by $f(x) - g(x)$. The total area of all the strips is then obviously $\int_{-5}^{5} ( f(x) - g(x) )\ dx$.