222 
CALCULUS 
EXERCISES 
1. Compute the line integral 
C 
when f=y-\-e x and C is the straight line joining the points (0, 1) 
and (1, 0). 
2. The same when f = 2x 2 + 3xy and C is the arc of the circle 
x 2 + y 1 — 1, which is contained in the first quadrant. 
3. Compute the line integral 
J P clx + Q cly 
when P = x 4- y, Q — xy, and C is the straight line of Question 1. 
(Two answers, corresponding to the two possible senses of C.) 
4. The same when C is the square whose sides lie along the co 
ordinate axes and the lines x = a, y = a. 
4. Green’s Theorem in Two Dimensions. Let P be a function of 
(x, y), continuous, together with dP/dy, within and on the boundary 
y of a region S. Form the double integral 
It can be evaluated by means of the iterated 
Fig. 51 integral, Chap. Ill, § 4 : 
S a Y 0 a a 
These latter integrals can be expressed in terms of the line 
integral, 
Ü 
extended over the complete boundary C of S. Since this last inte 
gral, when the sense of the description of C is reversed, goes over 
into the negative of its former value, it is important to be able to say 
which sense is intended. We say that C is described in the positive 
sense if, the region S being thought of as a pond, a man who walks