Calculus: Analytic Geometry and Calculus, with Vectors

13.3 Double integrals 671

Then show that

f f f (,,y) dS=

Joa dxJoy f (x,Y)dY ab dx foa

f (,,,Y)

dY

a

+fb dx fy_bf(x,y) dy.

3 Supposing that S is a bounded set having positivearea, interpret and prove

the statement

ISI=ffs1dS.

Hint: Look at the definition of double integrals.

(^4) Supposing that f is continuous over S, use Theorem 13.38 and the method
of Problem 1 to obtain iterated integrals equal to
ffsf(x,y) dS
when S is the set in E2 bounded by the graphs of the equations
(a) y = x2, y = mx + b (m,b > 0)
(b) y=x,y=2x,x=1
(c) y=0,yx,y=3x-2
(d) y=xs,y=x
(e) y=es,y0,x0,x=1
(f) y = x, y = sin x, x = 7r/2
5 Observe that sin (x + y) is continuous and nonnegative over the square
S in the xy plane having opposite vertices at the points (0,0) and (7r/2, it/2).
Evaluate
f fssin(x + y) dx dy.
,4ns.: 2.
6 Evaluate
1
ffgl+x+ydxdy
when Q is the square having opposite vertices at the points (0,0) and (1,1).
Ans.: 3 log 3 - 4 log 2.
7 Supposing that 0 < a < b and 0 < p < q, evaluate
ffs
eylx dS
when S is the region bounded by the lines having the equations x = a, x = b,
y = px, and y = qx. 11ns.: -ff(eQ - e')(b2 - a2).
8 Supposing that S is the square having opposite vertices at the points(0,0)
and (1,1), show that

If

1 df f 1^1

__f]- log (1-x)d.,

(1) s 1 - xy x dy = o dx 0 1 - -'Ydy o x