Question

Find the area bounded by the curves x = y^2 – 4y and x = 2y – y^2. Your work must include an integral in one variable

Answer 1

you can draw x = y^2 - 4y and x = 2y - y^2 on the same graph but make x the vertical axis for ease.
Find the roots...
x = y(y - 4) and x = y(2 - y)
so y = 0 or 4 for first one and 0 or 2



When you draw this you should see the graph of x = y(2 - y) lies 'above' the graph
of y(y-4) in the shaded part.

We always do *** Area of section = Area bounded by TOP curve and horizontal - Area bounded by BOTTOM curve and horizontal.

For a graph of f(t) above g(t) (i.e. f(t) > g(x) in the interval [a,b] this is \[\int\limits_{a}^{b}(f(t) - g(t)) dt\]

Of course replace t with whatever variable you require.