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Arc Length and Surface of Revolution Worksheet

Arc Length and Surface of Revolution Worksheet
  • Page 1
 1.  
Find the arc length of the curve y = 5 - 3 2x from x = 3 to x = 5.
a.
413 units
b.
213 units
c.
132 units
d.
13 units
e.
134 units


Solution:

y = 5 - 3 / 2x
[Write the equation.]

dydx = - 3 / 2
[Find dydx.]

Arc length of the curve = 351+(dydx)2 dx
[Formula for arc length.]

= 351+(-32)2 dx
[Substitute dydx = - 3 / 2.]

= 35132 dx

= 132 [ x ]35
[Evaluate the integral.]

= 132 (5 - 3) = 13

So, the arc length of the curve = 13 units


Correct answer : (4)
 2.  
Find the arc length of the curve y = 3 2(x)23 from x = 1 to x = 8.
a.
75 units
b.
32 units
c.
55-22 units
d.
5-2 units
e.
55+22 units


Solution:

y = 3 / 2x23
[Write the equation.]

dydx = 3 / 2 × 2 / 3 x-13 = x-13
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

= 181+x-23 dx
[Substitute dydx = x-13 .]

= 181+1x23 dx

= 18x23+1x13 dx

= [(1+x23)32 ]18
[Use substitution method to evaluate Integral.]

= 532-232 = 55-22

So, the arc length of the curve = 55-22 units


Correct answer : (3)
 3.  
Find the arc length of the curve y = 1 2x2 + 5 from x = 0 to x = 2.
a.
(1 2ln (2 + 5)) units
b.
(5 + 1 2ln (2 + 5)) units
c.
(ln (2 + 5)) units
d.
(5 + 2 ln (2 + 5)) units
e.
(5 + ln (2 + 5)) units


Solution:

y = 1 / 2x2 + 5
[Write the equation.]

dydx = x
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

= 021+x2 dx
[Substitute dydx = x.]

= [ x21+x2+12sin h-1 (x) ]02

= [1+4+12ln (2 + 22+1) - 0]
[Use sin h-1(x) = ln (x + x2+1) .]

= [5 + 1 / 2ln (2 + 5) ]

So, the arc length of the curve = (5 + 1 / 2ln (2 + 5)) units.


Correct answer : (2)
 4.  
Find the arc length of the curve y = x312+1x from x = 2 to x = 3.
a.
37 12units
b.
25 12units
c.
3 4units
d.
7 4units
e.
29 12units


Solution:

y = x312+1x
[Write the equation.]

dydx = x24-1x2 = x4-44x2
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

231+(x4-44x2)2 dx
[Substitute dydx = x4-44x2.]

2316x4+x8+16-8x416x4 dx

= 23(x4+4)216x4 dx

= 23x4+44x2 dx

= 23x24 dx + 23x-2 dx

= [x312]23 - [1x ]23

= 2712-812-13+12

= 1912-13+12 = 19-4+612

= 21 / 12 = 7 / 4
[Simplify.]

So, the arc length of the curve = 7 / 4units


Correct answer : (4)
 5.  
Find the arc length of the curve y = ln sin x from π6 to π3.
a.
0.767 units
b.
1.865 units
c.
0.549 units
d.
0 units
e.
1.316 units


Solution:

y = ln sin x
[Write the equation.]

dydx = 1sinx cos x = cot x
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

= π/6π/31+cot2x dx
[Substitute dydx = cot x.]

= π/6π/3 cosec x dx

= [ ln | tan (x2) | ]π/6π/3

= ln | tan (π6) | - ln | tan (π12) |

= ln (13) - ln (2 - 3)

= - 0.549 + 1.316

0.767

So, the length of the curve 0.767 units


Correct answer : (1)
 6.  
Find the arc length of the curve y = 1 + ln cos x from x = 0 to x = π4.
a.
0 units
b.
0.414 units
c.
1.414 units
d.
0.3464 units
e.
0.8814 units


Solution:

y = 1 + ln cos x
[Write the equation.]

dydx = 1cosx (- sin x) = - tan x
[Find dydx]

Arc length of the curve = 0π/41+tan2x dx
[Use formula for arc length.]

= 0π/4sec x dx
[Substitute dydx = - tan x.]

= [ ln | sec x + tan x | ]0π/4

= ln | 2 + 1 |

0.8814

So, the arc length of the curve 0.8814 units


Correct answer : (5)
 7.  
Find the arc length of the curve y = 2x32 + 3 from x = 0 to x = 9.
a.
[(82)32 - 1] units
b.
2 27 [(82)32 + 1] units
c.
2 27 (82)32 - 1 units
d.
2 27 [(82)32 - 1] units
e.
2 27 (82)32 + 1 units


Solution:

y = 2x32 + 3
[Write the equation.]

dydx = 3x12
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

= 091+9x dx
[Substitute dydx = 3x12.]

= 2 / 27 (1+9x)32 |09
[Evaluate the integral.]

= 2 / 27 (82)32-227

= 2 / 27 [(82)32 - 1]

So, the arc length of the curve = 2 / 27 [(82)32 - 1] units


Correct answer : (4)
 8.  
Find the arc length of the curve y = x48+14x2 from x = 1 to x = 2.
a.
35 16units
b.
37 16units
c.
39 16units
d.
15 8units
e.
33 16units


Solution:

y = x48+14x2
[Write the equation.]

dydx = x32-12x3 = x6-12x3
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

= 121+(x6-12x3)2 dx
[Substitute dydx = x6-12x3.]

= 124x6+x12-2x6+14x6 dx

= 12(x6+1)24x6 dx

= 12x6+12x3 dx

= 1 / 212x3 dx + 1 / 212x-3 dx

= [x48 ]12 - [14x2 ]12

= 168-18-116+14

= 32-2-1+416 = 33 / 16

So, the arc length of the curve = 33 / 16units


Correct answer : (5)
 9.  
Find the arc length of the curve y = x510+16x3 from x = 1 to x = 2.
a.
779 240units
b.
709 240units
c.
7 48units
d.
31 10units
e.
1 48units


Solution:

y = x510+16x3
[Write the equation.]

dydx = x42-12x4 = x8-12x4
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

= 121+(x8-12x4)2 dx
[Substitute dydx = x8-12x4.]

= 124x8+x16+1-2x84x8 dx

= 12(x8+1)24x8 dx

= 12x8+12x4 dx

= 1 / 212x4 dx + 1 / 212x-4 dx

= [x510 ]12 - [16x3 ]12

= 3210-110-148+16

= 3110+748 = 779 / 240

So, the arc length of the curve = 779 / 240units


Correct answer : (1)
 10.  
Find the arc length of the curve y = ex+e-x2 from x = 0 to x = 2.
a.
e4+1e2 units
b.
e4-12e2 units
c.
e4+12e2 units
d.
e4-1e2 units
e.
e4-12 units


Solution:

y = ex+e-x2
[Write the equation.]

dydx = ex-e-x2
[Find dydx.]

Arc length of the curve = ab1+(dydx)2 dx
[Formula for arc length.]

= 021+(ex-e-x2)2 dx
[Substitute dydx = ex-e-x2.]

= 024+e2x+e-2x-24 dx

= 02(ex+e-x)24 dx

= 02ex+e-x2 dx

= 1 / 2 [ex-e-x ]02

= 1 / 2 [e2-e-2-e0+e-0]

= 1 / 2 [e4-1e2] = e4-12e2

So, the arc length of the curve = e4-12e2 units


Correct answer : (2)

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