Calculus Assignment 14 and 15 Solutions

The article provides solutions to Calculus Assignment 14 and 15 problems. Assignment 14 includes integrals of functions, while Assignment 15 covers the Lorentz curve and coefficient of inequality. The Lorentz curve is used to measure income inequality in a population. The article also includes a real-life example of the Lorentz curve for the USA in 1968 and 2010.

Running head: CALCULUS 1
Calculus
Name
Institution

CALCULUS 2
Assignment 14
Question 1
∫
0
1
(3 x2 −2)4 x dx
Let 3 x2−2=u so that dx= du
6 x
∫(3 x2 −2)4 xdx=∫ (u )4 x du
6 x = 1
6 ∫(u)4 du= 1
30 (u)5
¿ 1
30 ( 3 x2−2)5
∫
0
1
(3 x2 −2)4 x dx= 1
30 [ (3 x2−2)5 ]0
1
= 1
30 { (3(1)2−2 )5
−(3 (0)2−2)5 }
¿ 1
30 { ( 1 )5 − (−2 )5 }= 1
30 (1+32 ) =33
30 =1.1

CALCULUS 3
Question 2
∫
1
e
3 x−1 dx
∫
1
e
3 x−1 dx=3∫
1
e
1
x dx= [ 3 ln (x ) ] 1
e
=3 ln ( e ) −3 ln ( 1 ) =3
Question 3
∫ 6 x
x2+5 dx
Let x2+5=u so that dx= du
2 x
∫ 6 x
x2+5 dx=∫ 6 x
u
du
2 x =∫ 3
u du=3∫ 1
u du=3 ln ( u )+C

CALCULUS 4
∫ 6 x
x2+5 dx=3 ln ( x2 +5 ) +C
Question 4
∫7 x √ 3 x2 +8 dx
Let 3 x2+ 8=u so that dx= du
6 x
∫7 x √3 x2 +8 dx=∫7 x √u du
6 x = 7
6 ∫ √u du= 7
6 ∫u
1
2 du
7
6 ∫u
1
2 du=7
6 ( 1
1
2 +1 ) u
1
2 +1
+C= 7
6 ( 2
3 ) u
3
2 +C=7
9 u
3
2 +C
7
9 u
3
2 +C= 7
9 (3 x2+ 8)
3
2 +C
∫7 x √3 x2 +8 dx= 7
9 (3 x2+8)
3
2 +C

CALCULUS 5
Assignment 15
Lorentz curve
Part a
The sketch of the Lorentz curve f ( x )= 19
20 x2 + 1
20 x using Desmos graphing calculator is shown
below.
Part b
f ( x )= 19
20 x2 + 1
20 x
f ( 0.60 )= 19
20 (0.60)2+ 1
20 ( 0.60 )=0.342+0.03=0.372
That is, when the population is 60 % the national income is 37.2 %
Part c

CALCULUS 6
The coefficient of inequality is the area bounded by the Lorentz curve and the line f ( x )=x from
x=0 ¿ x=1 as shown in shaded region of the figure below.
The equation of the shaded region is ( fx ) =x− ( 19
20 x2+ 1
20 x )=x −19
20 x2 − 1
20 x
Coefficient of inequality=Area of shaded region=∫
0
1
( x− 19
20 x2− 1
20 x ) dx
∫
0
1
( x− 19
20 x2− 1
20 x ) dx= [ 1
2 x2 −19
20 ( 3 ) x3− 1
20 ( 1
2 ) x
2
] 0
1
¿ [ 1
2 x2− 19
60 x3− 1
40 x2
]0
1
¿ 1
2 (1)2− 19
60 ( 1 ) 3− 1
40 ( 1 ) 2= 19
120
Part d

CALCULUS 7
Figure 1 below shows a real life example of Lorentz curve for USA in the year 1968 and 2010.
The graph shows that there has been an increase in income inequality in the US since 1968.
Figure 1: Lorentz curve (http://mathscinotes.com/2015/01/income-inequality-and-the-gini-
coefficient/)