Determine the area of a triangle by integration...under the line y=x+2

tooma

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[FONT=&quot]I am trying to find by the integration the area enclosed by the x-axes, y-axes and is under the line y=x+2 (i.e, a triangle in the second quarter). When I take a horizontal element, the final result for the integration is -2. When I take a vertical element, the final result of the integration is +2. Is it possible to have two different signs for the results, if yes, why? Can you please solve it with the two elements types if my final result(s) are wrong.[/FONT]
 
A vertical method would involve "top curve minus bottom curve."

You're in Quadrant II.

x goes from -2 to 0.

The top curve is y = x + 2.

The bottom curve is the x-axis, which is y = 0.

The integrand is [(x + 2) - 0] = [x + 2].

Integrate (x + 2)dx from x = - 2 to x = 0.


- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -


A horizontal method would involve "right curve minus left curve."

Again, you're still in Quadrant II.

y goes from 0 to 2.

The right curve is x = 0.

y = x + 2 =>

x = y - 2

So, the left curve is x = y - 2.

The integrand is [0 - (y - 2)] = [- y + 2].


Integrate (- y + 2)dy from y = 0 to y = 2.

_____________________________________


They both should equal +2.
 
Last edited:
A vertical method would involve "top curve minus bottom curve."

You're in Quadrant II.

x goes from -2 to 0.

The top curve is y = x + 2.

The bottom curve is the x-axis, which is y = 0.

The integrand is [(x + 2) - 0] = [x + 2].

Integrate (x + 2)dx from x = - 2 to x = 0.


- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -


A horizontal method would involve "right curve minus left curve."

Again, you're still in Quadrant II.

y goes from 0 to 2.

The right curve is x = 0.

y = x + 2 =>

x = y - 2

So, the left curve is x = y - 2.

The integrand is [0 - (y - 2)] = [- y + 2].


Integrate (- y + 2)dy from y = 0 to y = 2.

_____________________________________


They both should equal +2.


Now I understand it. Thanks a lot
 
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