Answer:
The midpoint of the segment is (-0.5,0.5).
Step-by-step explanation:
Midpoint of a segment:
The midpoint of a segment is given by the mean of its coordinates.
Segment with end points (4,2),(-5,-1)
x-coordinates: 4 and -5
y-coordinates: 2 and -1
x-coordinate of the midpoint:
y-coordinate of the midpoint:
The midpoint of the segment is (-0.5,0.5).
Rewrite the equations of the given boundary lines:
<em>y</em> = -<em>x</em> + 1 ==> <em>x</em> + <em>y</em> = 1
<em>y</em> = -<em>x</em> + 4 ==> <em>x</em> + <em>y</em> = 4
<em>y</em> = 2<em>x</em> + 2 ==> -2<em>x</em> + <em>y</em> = 2
<em>y</em> = 2<em>x</em> + 5 ==> -2<em>x</em> + <em>y</em> = 5
This tells us the parallelogram in the <em>x</em>-<em>y</em> plane corresponds to the rectangle in the <em>u</em>-<em>v</em> plane with 1 ≤ <em>u</em> ≤ 4 and 2 ≤ <em>v</em> ≤ 5.
Compute the Jacobian determinant for this change of coordinates:
Rewrite the integrand:
The integral is then