Thomas Calculus 13th [Solutions]

1110 Chapter 15 Multiple Integrals
48. The region R is shaded in the graph below.
( 1, 1)
2
y
1
2
0 2
x
As ranges from 3 /4 to 5 /4 the ray at angle enters R at r sec and leaves R at r 2. Thus
R
2 5 /4
2 2 2 4
x y dA r r dr d
3 /4 sec
5 /4 2
5 /4 5 /4
1 2 1 2 1 5 /4
1 1
r d 2cos 1 d cos2 d sin 2
3 /4
3 /4
3 2 4 4
sec
3 8 4
49-52. Example CAS commands:
Maple:
f : (x,y) - y/(x^2+y^2);
a,b : 0,1;
f1: x - x;
f2 : x - 1;
plot3d( f(x,y), y f1(x)..f2(x), x a..b, axes boxed, style patchnogrid, shading zhue, orientation [0,180],
title "#49(a) (Section 15.4" );
# (a)
q1: eval( x a, [x r*cos(theta),y r*sin(theta)] ); # (b)
q2 : eval( x b, [x r*cos(theta),y r*sin(theta)] );
q3 : eval( y f1(x), [x r*cos(theta),y r*sin(theta)] );
q4 : eval( y f2(x), [x r*cos(theta),y r*sin(theta)] );
theta1: solve( q3, theta );
theta2 : solve( q1, theta );
r1: 0;
r2 : solve( q4, r );
plot3d(0,r r1..r2, theta theta1..theta2, axes boxed, style patchnogrid, shading zhue, orientation [-90,0],
title "#49(c) (Section 15.4)" );
fP : simplify(eval( f(x,y), [x r*cos(theta),y r*sin(theta)] ));
q5 : Int( Int( fP*r, r r1..r2 ), theta theta1..theta2 );
value( q5 );
# (d)
Mathematica: (functions and bounds will vary)
For 49 and 50, begin by drawing the region of integration with the FilledPlot command.
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