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

Review exercises 27 901
REVIEWEXERCISES27
1 Find ∫ C (2x −y)dx +xydy along the straightline
joining (−1, −1)and (1,1).
2 Find ∫ C (2x+y)dx+y3 dy
(a)along the curvey =x 3 between (1,1)and (2,8),
and
(b)along the straightline joining these points.
3 IfF = 3xyi +2e x yjfind ∫ CF ·ds whereC is the
straightliney = 2x +1 between (1,3)and (4,9).
4 ShowthatthefieldF = (2x +1)yi + (x 2 +x+1)jis
conservative and findasuitable potential function φ.
5 Find ∮ C (x2 i +4xyj) ·ds whereC is aclosed pathin
the form ofatriangle with vertices at
(0,0),(3,0)and(3,5).
6 Find ∫ y=3
y=0
7 Find ∫ y=2
y=0
∫ x=2
x=1 (3x −5y)dxdy.
∫ x=1+y
x=0 (2y +5x)dxdy.
8 Find
∫ x=2 ∫ y=1 ∫ z=3
x=0 y=0 z=0 (x2 +y 2 +z 2 )dzdydx.
9 Evaluate ∫∫ R (4y +x2 )dxdy whereRisthe interior of
the squarewith verticesat (0,0),
(4,4),(0,4)and(4,0).
10 Evaluate ∮ C ey dx +e x dywhereC isthe boundary
ofthe triangle formedbythe linesy =x,y = 5
andx = 0. Byconverting thisline integral intoa
doubleintegral verifyGreen’s theoremin the
plane.
11 TheregionRis bounded bytheyaxisand the lines
y=xandy=3−2x.
(a) Sketch the regionR.
(b) Find the volume between the surfacez =xy +1
and the regionR.
(a) Evaluate ∮ CF ·dswhereC is the curve enclosing
the regionR.
(b) Verify Green’stheorem in the plane.
(c) Thesurface,z(x,y),isgivenbyz = 1 +x+xy.
Calculatethevolumeunderthesurfaceandabove
the regionR.
13 The regionRisbounded bythexaxisand the curve
y=8+2x−x 2 .
(a) SketchR.
(b) Evaluate ∫∫ R 3x+2ydxdy
14 Evaluate
∫ 0 ∫ 4
(a) −1 3
∫ 3xydxdy
3 ∫ 2
(b) 2 0
∫ 4+x−ydydx
1 ∫ 4
(c) 0 2y x2 +y 2 dxdy
∫ −1 ∫ x 3
(d) −2 0 1dydx
15 Sketch the regionsofintegration ofthe double
integrals in Question14.
16 Ifa,b,candd are constantsshow that
∫ d ∫ b
f(x)g(y)dxdy
c a
is identical to
[∫ b
f(x)dx
] [∫ d ]
g(y)dy
a c
y
9
C
R
y = 9 – x 2
12 TheregionRis shown in Figure 27.27.
Thevector fieldFisgiven by
Solutions
F =y 2 i+3xyj
0
3
Figure27.27
The regionRforQuestion12.
x
1
2
3
2 (a) 1030.5 (b) 1031.25
3 1666.37
4 φ=x 2 y+yx+y+c