Exercise
Find the area of the region bounded by the graphs of the equations:
y-4x=0, x+y=5, y=0
Final Answer
Solution
First, we find out how the area looks like:
y-4x=0\Longrightarrow y=4x
x+y=5\Longrightarrow y=-x+5
4x=-x+5
5x=5
x=1
y-4x=0
0-4x=0
4x=0
x=0
In the equation y=0 we set
x+y=5
and get
x+0=5
x=5
The area looks like this:
Hence, it is a sum of two disjoint areas:
We will calculate them seperately:
S=S_1+S_2
The first area:
S_1=\int_0^1 4x dx=
=[4\cdot\frac{x^2}{2}]_0^1=
=[2x^2]_0^1=
=2\cdot 1^2-2\cdot 0^2=
=2-0=
=2
The second area:
S_2=\int_1^5 -x+5 dx=
= [-\frac{x^2}{2}+5x]_1^5=
=-\frac{5^2}{2}+5\cdot 5-(-\frac{1^2}{2}+5\cdot 1)=
=-\frac{25}{2}+25+\frac{1}{2}-5=
=20-12\frac{1}{2}+\frac{1}{2}=
=8
Hence, we got
S_2=8
Lastly, we sum up the results:
S=S_1+S_2=
=2+8=10
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