# Thread: How do I find the limits of integration?

1. ## How do I find the limits of integration?

In a problem such as this:

Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
y + x = 5,
y - x = 0, y + 3x = 4

or this:

Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
x = y3 - 18y,
y + 6x = 0
How do I find the limits of integration?

2. Originally Posted by bigdan
In a problem such as this:

Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
y + x = 5,
y - x = 0, y + 3x = 4

or this:

Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
x = y3 - 18y,
y + 6x = 0
How do I find the limits of integration?
first of all, draw your self those lines and you will see limits for integrations ...

to draw any line you need just 2 points and connect them ... (for curves you need at least 3 points)

so ....
$\displaystyle x+y = 5 \Rightarrow x = 5-y$ so for y = 0 you have x = 5 , and for let's say y = 5 you will have x = 0 so you have 2 points M(5,0) N(0,5) yo you draw the line through those 2 points ... do the same thing with another 2 equations and you will get that region that you need to integrate

to see points of the intersecting, just put first equation in second and see where those 2 lines intersect , same thing with second and third and of course first and third .... and from that you will get your points of integration...

intersecting point of the first and second
$\displaystyle y+x = 5 \Rightarrow y = 5-x$ put in the second $\displaystyle y-x = 0 == 5-x-x=0 \Rightarrow 2x = 5 \Rightarrow x = \frac {5}{2}$
intersecting first and third
$\displaystyle y+x =5 \Rightarrow y=5-x$ put in the third $\displaystyle y+3x=4 == 5-x+3x=4 \Rightarrow 2x=-1 \Rightarrow x=- \frac {1}{2}$
and second and third ... x =1

Hint: There are two regions of integration.... calculate both and add them $\displaystyle \displaystyle \int _? ^? dx \int _? ^? dy + \int _? ^? dx \int_? ^? dy$

P.S. second problem do the same way

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# how to take limit in integration

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