Future Engineering Mathematics: Line Integrals, Double Integrals, and Power Method

2018
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FUTURE ENGINEERING MATHEMATICS
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QUESTION 1
Part a (2mks)
Sketch the graphs of y=2sin2(x ) and y=2cos2 (x) from x=−π ¿ x=π
-4 -3 -2 -1 0 1 2 3 4
x
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Y=2sin2(x)
-4 -3 -2 -1 0 1 2 3 4
x
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Y=2cos2(x)

Part b
Using a double integral, calculate the area enclosed by the curves in the interval x=0 ¿ x= π
4
∫∫
0
π
4
y dy=∫∫
0
π
4
2 sin2 ( x ) dx
Computing the indefinite integral
Computing the boundaries:
¿ π
4 − 1
2 −0
Simplifying further,
¿ π
4 − 1
2
∫
0
π
4
( π
4 − 1
2 )dx= π2−2 π
16 ≈ 0.22415
Part c
Determine area enclosed by the curves in the interval

∫
π
4
π
2
( 2sin2 x−2 cos2 x ) dx
Rewrite or simplifying the function,
∫−2 cos 2 x dx
Substitute u=2 x → dx=1
2 du
−∫cos (u)du
Now solving:
∫cos ( u ) du
This is a standard integral,
¿−sin ( u )
Redoing the substitution,
¿−sin ( 2 x )
The problem is solved as,
∫−2 cos 2 x dx=−sin 2 x+C
¿ 1
Part d
Infinitesimal elements of volume are expressed in Cartesian coordinates by dV =dxdydz
∫∫∫dv =∫∫∫ d xdydz