Question

area between two curves...

Answer 1

How do you know when to use dy or dx?

Answer 2

And what's this?

Answer 3

Specifically, the picture

Answer 4

after you sketch the curve, you draw in a thin rectangle
and figure out the bounds of that rectangle.

in this problem, a vertical rectangle does not look convenient:
1) the limits (top and bottom) are the same curve
you could figure it out, but it looks complicated
2) the bounds change at x=0, (from the red curve to the blue one)
that means you must divide the problem into two separate integrals

Answer 5

In this problem, it makes sense to use a horizontal rectangle
it will have width \( x_2 - x_1 \) and height dy
Here \(x_2\) is the "right boundary", the blue curve and \( x_1\) is the red curve
(notice we do "big x" - "smaller x" so that we get a positive number for the width)
so the area of one rectangle is
\[ (x_2 - x_1 ) \ dy \]
because we have a height dy, we want all the variables in terms of y
That is, replace the x's with their definitions:
\[ 2-y^2 -y^4 \ dy \]