# Thread: Areas of Common Geometric Figures homework help

1. ## 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???

2. ## Re: Areas of Common Geometric Figures homework help

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

3. ## Re: Areas of Common Geometric Figures homework help

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?

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

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?

4. ## 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....

5. ## Re: Areas of Common Geometric Figures homework help

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

Code:
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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.)

6. ## Re: Areas of Common Geometric Figures homework help

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