082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

888 Chapter 27 Line integrals and multiple integrals
EXERCISES27.6
1 Evaluate
∫ 1 ∫ 3
(a) z 2 dxdz
0 0
∫ 2 ∫ 3
(c) x 2 y+1dxdy
1 2
∫ 4 ∫ 2
(b) xdydz
0 0
∫ 1 ∫ 3 y
(d)
−1 2 x dxdy
2 R isthe shaded region shown in Figure27.16.
Evaluate ∫∫ R x + √ ydxdy
(a) performing the integration with respect toxfirst,
(b) performing the integration with respect toyfirst.
y
4
2
0 1
y = x 2
2 x
Figure27.16
Theregion ofintegration forQuestion2.
3 Evaluate ∫∫ R (5x2 +2y 2 )dxdywhereRis the interior
ofthe triangularregion boundedby
A(1,1),B(2, 0)andC(2, 2).
4 (a) Sketch the region ofintegration ofthe double
integral
∫ 3 ∫ √ 4−y
ydxdy
1 1
(b) Evaluate the integral byfirst reversingthe order
ofintegration.
5 Evaluate
∫ 1 ∫ 5
(x 2 +y 2 )dxdy
−1 1
6 Evaluate
∫ ∫
(x 2 +y 2 )dxdy
R
over the triangularregionRwith vertices at
(0,0),(2,0)and(1,1).
7 Evaluate
∫
(x +2y) −1/2 dxdy
R
over the regionRgiven byx −2y 1 and
xy 2 +1.
8 Evaluate ∮ C (sinx +cosy)dx +4ex dywhereC isthe
boundary ofthe triangle formedby the points(1,0),
(3,0)and(3,2).Byconverting thisline integralinto a
doubleintegral, verifyGreen’stheorem.
9 Evaluate the line integral
∮
(x+y)dx+3xydy
C
whereC is the boundary ofthe triangle formedbythe
points(0,0),(2,0)and(0,5).Expressthelineintegral
in terms ofan appropriatedoubleintegral and
evaluate this.Verify Green’stheorem.
Solutions
1 (a) 1 (b) 8x (c) 10.5 (d) 0
2 (a)
3
33
2
20
3
(b)
20
3
7
2
.TheregionRis shown in FigureS.26.
3
y
2
x = y 2 + 1
y = –
x 1
– –
1
2 2
4 (b) 1.35435
5
6
256
3
4
3
0 1 2 3 4 5
FigureS.26
8 92.306
9 20
x