# Areas of Common Geometric Figures homework help

• Dec 14th 2012, 01:07 AM
asilvester635
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???
• Dec 14th 2012, 01:16 AM
MarkFL
Re: Areas of Common Geometric Figures homework help
No, they would be different since the integrand is different in each case.
• Dec 14th 2012, 06:24 AM
Soroban
Re: Areas of Common Geometric Figures homework help
Hello, asilvester635!

Quote:

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?

$\displaystyle (e)\;\int^6_{\text{-}4} |f(x)|\,dx$

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.

$\displaystyle (f)\;\int^6_{\text{-}4}[f(x)+2]\,dx$

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?
• Dec 14th 2012, 09:31 AM
asilvester635
Re: Areas of Common Geometric Figures homework help
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....
• Dec 14th 2012, 12:40 PM
Soroban
Re: Areas of Common Geometric Figures homework help
Hello, asilvester635!

Quote:

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: $\displaystyle y \:=\:2x-4$ and the coordinate axes.

Code:

        |         |      /     ----+-----*----         |::::/2         |:::/         |::/         |:/         |/         *       /|         |
The integral is: .$\displaystyle A \;=\;\int^2_0(2x-4)\,dx$

We have: .$\displaystyle x^2 - 4x\,\bigg|^2_0 \;=\;(4-8) - (0-0) \;=\;{\color{red}-}4$ . (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.)
• Dec 14th 2012, 01:56 PM
skeeter
Re: Areas of Common Geometric Figures homework help
Quote:

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