Grigori S. answered • 06/15/13
Certified Physics and Math Teacher G.S.
Let's find the equation for the segment "c". It gives you values of the function y=y(x) and helps you to rewrite your integral in terms of "x" only. Using two given points (1,0) and (3,1) we can find that the slope of the segment is
m = (y2-y1)/(x2-x1) = (1-0)/(3-1) = 1/2
You can write the equation of the segment in generic slpe- intercept form:
y = mx +b
Because y = 0 if x = 1, we have: b = -1/2 and the eqaution is: y = (x/2) - 1/2.
Thus
x-y = (1/2)(x+1) and xy = (1/2)x(x-1) and dy = (1/2)dx
Your integral (with x=1 for the lowest limit and x=3 for the highest limit)
∫ xydx + (x-y)dy can be reduced to
(1/2)∫ (x2 +1)dx = 23/6
Answer: intehral = 23/6
Grigori S.
aYou can also start with parametrization of a line, such as taking
x = 1 +at and y = ßt, where 0 = t = 1. Because x = 1 and 3, and y = 0, 1
we have: x = 1+2t, y = t . Thus dx = 2dt, dy = dt. Substituting these expressions into your original integral we will obtain:
?xydx + ?(x-y)dy = 2?t(1+2t)dt + ?(1+t)dt = 23/6
with the limits of integration for t shown above.
06/15/13