2014-test-02

Name:Student ID:Major:
Answer all the questions(20marks each).Write your answer on A4papers provided, with your ID and name.Hand in this Question Paper with your answer. Second Mid-Term of Matb210Engineering Mathematics I(A)
8:00-9:30p.m.17/12/2014E22-1016and E22-1013)
12345
Ans.
1.Evaluate the triple integral ∫∫∫
D
2yzdV,where D is the solid tetrahedron P ABC
with vertices A(0,0,0),B(0,1,0),C(2,1,0)and P(0,1,2).
2.Evaluate the triple integral ∫∫∫
D
zdV,where
D={(x,y,z)|x2+y2+z2≤a2,z≥0}is the upper-hemisphere.
3.Find the average value of the function f(x,y)=4xye x2y over the rectangle
R={(x,y)|0≤x≤2,0≤y≤3}.
4.Find the area of the region R that is enclosed by one loop of polar curve
r=2sin(4θ).Hint:sin2θ=1−cos2θ
2
.
5.(a)Draw a picture of the region of integration of
∫1
0∫2x
x
dydx.
(b)Exchange the order of integration to express the integral in part(a)in terms
of integration in the order :Student ID:Major:
Answer all the questions(20marks each).Write your answer on A4papers provided, with your ID and name.Hand in this Question Paper with your answer. Second Mid-Term of Matb210Engineering Mathematics I(D)
8:00-9:30p.m.17/12/2014E22-1016and E22-1013)
12345
Ans.
1.(a)Draw a picture of the region of integration of
∫1
∫2x
x
dydx.
(b)Exchange the order of integration to express the integral in part(a)in terms
of integration in the order dxdy.
2.Evaluate the triple integral
∫∫∫
D
2yzdV,where D is the solid tetrahedron P ABC with vertices A(0,0,0),B(0,1,0),C(2,1,0)and P(0,1,2).
3.Find the area of the region R that is enclosed by one loop of polar curve
r=2sin(4θ).Hint:sin2θ=1−cos2θ
2
.
4.Evaluate the triple integral
∫∫∫
D
zdV,where
D={(x,y,z)|x2+y2+z2≤a2,z≥0}is the upper-hemisphere.
5.Find the average value of the function f(x,y)=4xye xy2over the rectangle
R={(x,y)|0≤x≤3,0≤y≤2}.
Name:Student ID:Major:
Answer all the questions (20marks each).Write your answer on A4papers provided,with your ID and name.Hand in this Question Paper with your answer.Second Mid-Term of Matb 210Engineering Mathematics I (A)8:00-9:30p.m.17/12/2014E22-1016and E22-1013)
1.Evaluate the triple integral ∫∫∫
D
2yzdV,where D is the solid tetrahedron P ABC with vertices A (0,0,0),B (0,1,0),C (2,1,0)and P (0,1,2).
Solution .∫∫∫D 2yzdV =∫10∫2y 0∫2y −x
02yz dz dy,dx =8
15.
2.Evaluate the triple integral ∫∫∫
D
zdV,where D ={(x,y,z )|x 2+y 2+z 2≤a 2,z ≥0}is the upper-hemisphere.Solution .Rewrite the solid region D as D ={(x,y,z )|−a ≤x ≤a,−√a −x 2≤y ≤√
a 2−x 2,0≤z ≤√
a 2−x 2−y 2}.Then ∫∫∫
D zdV =∫a −a ∫√a −x 2−√
a −x 2∫√a 2−x 2−y 20z dz dy dx =∫a −a ∫√a −x 2−√a −x 2
[z
22]√
a 2220dy dx =12∫a −a ∫√a −x 2
−√a −x 2(a 2−x 2−y 2)dy dx =12∫a −a [(a 2−x 2)y +y 33]√
a −x 2−√a −x 2dx =43∫
a −a (a 2−x 2)3/2dx ∗=4
3∫0π(a 2−(a cos θ)2)3/2d (a cos θ)=−43∫0πa 4sin 4θdθ=4a 43∫π
sin 4θdθ=4a 43∫π0(1−sin 2θ2)2dθ=4a 43∫π
01−2sin 2θ+sin 2(2θ)
4dθ
=a 43∫π01+1−sin(4θ)2dθ=,3.Find the average value of the function f (x,y )=4xye x 2
y over the rectangle R ={(x,y )|0≤x ≤2,0≤y ≤3}.Hint:Average of f over region R is ∫∫
R
f (x,y )dA ∫∫R
1dA .Solution .We first evaluate the double integral of f (x,y )over the region R as follows:∫∫R f (x,y )dA =
∫3
0∫2
04xye x 2y
dxdy =∫3
∫2
02e x 2
y d (x 2y )dy =∫30[2e x 2y ]x =2x =0dy =2∫30(e 4y −1)dy =2[e
4y 4−y
]30=2[e 12
−1
4−3]=e 12
−132
,and
the area of the region R is
∫∫
R
1dA =2×3=6.
The average value of f (x,y )over R is 16∫∫
R
f (x,y )dA =e 12−13
12.
4.Find the area of the region R that is enclosed by one loop of polar curve r =
2sin(4θ).Hint:sin 2θ=1−cos 2θ
2
.Solution .Area of R =∫π/40∫r =r max (θ)r =r min (θ)1·r drdθ=∫π/40∫2sin(4θ)
r drdθ=12∫π/40(2sin 4θ))2dθ==2∫π/40sin 24θdθ=2∫π/40
1−cos 8θ2dθ=∫π/4
0(1−cos 8θ)dθ=π4.
5.(a)Draw a picture of the region of integration of ∫10
∫2x
x
dydx.(b)Exchange the order of integration to express the integral in part (a)in terms
of integration in the order dxdy.
Solution .(a)
(b)∫10
∫2x x
dydx =
∫1
∫
x max (y )=y
x min (y )=y/2
dxdy +
∫
2
1
∫
x max (y )=1
x min (y )=y/2
dxdy.。