Thread: Areas of Common Geometric Figures homework help

wouldn't letters d, e, and f be the same answer??? since your taking the area over the same interval???

No, they would be different since the integrand is different in each case.

Hello, asilvester635!

Wouldn't letters d, e, and f be the same answer?
. . Certainly not!

Did you realize that some answers are negative?
Didn't you make any sketches?




All the regions below the x-axis are reflected upward!

Code:

                           2|
                        ..* * *..               *
                      *:::::|:::::*           *:::*
      *..           *:::::::|:::::::*       *:::::::*
      ::::*..      *::::::::|::::::::*    *:::::::::::*
      ::::::::*.  ::::::::::|:::::::::: *:::::::::::::::*
  - - + - - - - - * - - - - * - - - - * - - - - + - - - - * - -
     -4          -2         |         2         4         6
                            |

We want the area of the shaded region.





The entire graph is moved 2 units upward!

Code:

                            |
                           4+                   *
                            |                 *:::*
                            |               *:::::::*
                            |             *:::::::::::*
                            |           *:::::::::::::::*
                 .*        2+         *:::::::::::::::::::*
             .*:::|         |         |:::::::::::::::::::|
         .*:::::::|*        |        *|:::::::::::::::::::|
      *:::::::::::|:*.      |      .*:|:::::::::::::::::::|
      |:::::::::::|:::*..   |   ..*:::|:::::::::::::::::::|
  - - + - - - - - + - - - * * * - - - + - - - - + - - - - + - -
     -4          -2         |         2         4         6
                            |

We want the area under the graph.

Got it?

holy crap... yes i got it.... do we minus the area if it's below the x-axis??? like for the small triangle's area under the x-axis is 1 and he semicircle's area is 2pi and that big triangle above the x-axis area is 4....

Hello, asilvester635!

Do we subtract the area if it's below the x-axis?

Yes . . . and that is the "shortcut" way of looking at it.
Actually, since you are working with areas,
. . you should have been aware of this long ago.


Consider the area bounded by: and the coordinate axes.

Code:

        |
        |      /
    ----+-----*----
        |::::/2
        |:::/
        |::/
        |:/
        |/
        *
       /|
        |

The integral is: .

We have: . . (negative four)


If the region is below the x-axis, the value of the integral is negative.
. . (You can call it "negative area" if you like.)

Originally Posted by asilvester635

... do we minus the area if it's below the x-axis???

only if the lower limit of integration < upper limit of integration