Sketch the region enclosed by the given curves and find the area.

Question

Sketch the region enclosed by the given curves and find the area.

1. y=12-x², y=x²-6

2. x= 2y², x=4+y

3. y=1/x, y=x , y= 1/4 * x, x>0

Answer 1

desmos.com/calculator/db2scgviv3

Answer 2

If you are allowed to use a graphic display calculator, then you can get the sketch from there. Otherwise you can make a table of values and plot the coordinates.

The sketches are important because they can get you to think of alternative solutions. For an example, I have made a quick animation of #1. If you look at it you will see that if you shift both functions down 3 units, and because of symmetry, you can reduce this to a simple problem.

Paste this link into your browser to see the animation:

https://drive.google.com/file/d/1GLV_yUuuWyLa0BtUlbacna3dAsYsSw8V/view?usp=sharing

On number 2, notice that if you rotate everything 90 degrees counterclockwise you will have an easier problem with the same area. You basically look at the inverses of each function:

y = 2x² and y = x + 4, fairly easy integration again.

On number 3, again the sketch will lead you to a strategy. The sketch is the key!