1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
olchik [2.2K]
3 years ago
12

Evaluate integral _C x ds, where C is

Mathematics
1 answer:
borishaifa [10]3 years ago
5 0

Answer:

a.    \mathbf{36 \sqrt{5}}

b.   \mathbf{ \dfrac{1}{108} [ 145 \sqrt{145} - 1]}}

Step-by-step explanation:

Evaluate integral _C x ds  where C is

a. the straight line segment x = t, y = t/2, from (0, 0) to (12, 6)

i . e

\int  \limits _c \ x  \ ds

where;

x = t   , y = t/2

the derivative of x with respect to t is:

\dfrac{dx}{dt}= 1

the derivative of y with respect to t is:

\dfrac{dy}{dt}= \dfrac{1}{2}

and t varies from 0 to 12.

we all know that:

ds=\sqrt{ (\dfrac{dx}{dt})^2 + ( \dfrac{dy}{dt} )^2}} \  \ dt

∴

\int \limits _c  \ x \ ds = \int \limits ^{12}_{t=0} \ t \ \sqrt{1+(\dfrac{1}{2})^2} \ dt

= \int \limits ^{12}_{0} \  \dfrac{\sqrt{5}}{2}(\dfrac{t^2}{2})  \ dt

= \dfrac{\sqrt{5}}{2} \ \ [\dfrac{t^2}{2}]^{12}_0

= \dfrac{\sqrt{5}}{4}\times 144

= \mathbf{36 \sqrt{5}}

b. the parabolic curve x = t, y = 3t^2, from (0, 0) to (2, 12)

Given that:

x = t  ; y = 3t²

the derivative of  x with respect to t is:

\dfrac{dx}{dt}= 1

the derivative of y with respect to t is:

\dfrac{dy}{dt} = 6t

ds = \sqrt{1+36 \ t^2} \ dt

Hence; the  integral _C x ds is:

\int \limits _c \ x \  ds = \int \limits _0 \ t \ \sqrt{1+36 \ t^2} \  dt

Let consider u to be equal to  1 + 36t²

1 + 36t² = u

Then, the differential of t with respect to u is :

76 tdt = du

tdt = \dfrac{du}{76}

The upper limit of the integral is = 1 + 36× 2² = 1 + 36×4= 145

Thus;

\int \limits _c \ x \  ds = \int \limits _0 \ t \ \sqrt{1+36 \ t^2} \  dt

\mathtt{= \int \limits ^{145}_{0}  \sqrt{u} \  \dfrac{1}{72} \ du}

= \dfrac{1}{72} \times \dfrac{2}{3} \begin {pmatrix} u^{3/2} \end {pmatrix} ^{145}_{1}

\mathtt{= \dfrac{2}{216} [ 145 \sqrt{145} - 1]}

\mathbf{= \dfrac{1}{108} [ 145 \sqrt{145} - 1]}}

You might be interested in
60°<br> 18<br> Solve for x. Round to the nearest<br> tenth.
weeeeeb [17]

Answer:

You need to include a picture or add more o the question because I can't solve it without the rest of the problem. All I know is 60 degrees and 18. But what is the 18?

Step-by-step explanation:

5 0
3 years ago
The temperature, in degrees fahrenheit (°
vova2212 [387]
The temperature decreased by 5 degrees per hour, as -25 / 5 is -5.
5 0
3 years ago
Can someone please help ASAP?!!!
Lostsunrise [7]

Answer:

Step-by-step explanation:

4 0
3 years ago
colin has an ice cube tray with 12 identical compartments. Each compartment is a prism that is 4 cm. long, 3 cm. wide, and 3 cm.
a_sh-v [17]
Volume = length x width x height
V = 4 x 3 x 3
4 x 3 = 12
V = 12 x 3
V = 36 cubic cm.
1 cubic cm. = 1 ml.
36 cubic cm. x 12 compartments = 432 cubic cm.
432 cubic cm. = 432 cubic ml.
5 0
3 years ago
The cost of a cd is $18.95 makeup is 70%
kykrilka [37]
$5.69 Discount $13.27
8 0
3 years ago
Other questions:
  • What is the mathematical expression for “the $100 earned was split between all the people and each got $12.50”?
    12·2 answers
  • QUESTION 10
    5·1 answer
  • What is the slope of the line?
    5·1 answer
  • I cant fail this it’s my exam please help anyone
    14·2 answers
  • 12. A quality control procedure at a manufacturing facility involves selecting 5 items at random from a large batch, and then ac
    9·1 answer
  • What is the value of the expression below? 2³(3³ – 6) +4 A. 214 B. 172 C. 28 D. 200​
    7·2 answers
  • Which situation can be represented by y=12x?
    6·1 answer
  • Math help pls pls help urgent
    11·1 answer
  • If the graph passes the horizontal-line test, then the function is one-to-one. Functions that are one-to-one have inverses that
    15·1 answer
  • A room is three times as long as it is broad and it's height s 4.5 m. if the cost of carpeting its floor at rs 240 per m square
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!