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
Find the values of P for which the quadratic equation 4x²+px+3=0?​
klasskru [66]
<h3><u>Correct Questions :- </u></h3>

Find the values of P for which the quadratic equation 4x²+px+3=0 , provided that roots are equal or discriminant is zero .

<h3><u>Solution</u>:- </h3>

Let us Consider a quadratic equation αx² + βx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

For equal roots

  • D = 0

\quad\green{ \underline { \boxed{ \sf{Discriminant, D = β² - 4αc}}}}

So,

\sf{  β² - 4αc = 0}

Here,

  • α = 4
  • β = p
  • c = 3

Now,

\begin{gathered}\implies\quad \sf  p²- 4 \times 4 \times 3 =0 \end{gathered}

\begin{gathered}\implies\quad \sf  p²- 48 =0 \end{gathered}

\begin{gathered}\implies\quad \sf  p²=48\end{gathered}

\begin{gathered}\implies\quad \sf  p=±\sqrt{48}\end{gathered}

\begin{gathered}\implies\quad \sf  p=±\sqrt{2 \times 2 \times 2 \times 2 \times 3} \end{gathered}

\begin{gathered}\implies\quad \sf  p=± 2\times 2\sqrt{ 3 }\end{gathered}

\begin{gathered}\implies\quad \boxed{\sf{p=±4\sqrt{ 3 }}}\end{gathered}

Thus, the values of P for which the quadratic equation 4x²+px+3=0 are-

4√3 and -4√3.

7 0
2 years ago
In a parallelogram a diagonal of the length 20 cm is perpendicular to one of the sides. Find the longer side of parallelogram if
riadik2000 [5.3K]
All the sides are 20 cm resulting in the perimeter being 80
7 0
3 years ago
Read 2 more answers
Consider the following functions
TiliK225 [7]

Step-by-step explanation:

g(5)=3(5)-11

=15-11

=4

f(g(5))=2(4)²+4(4)+10

=2(16)+16+10

=32+26

=58

5 0
3 years ago
Divide: 4/15÷8/25. What is the answer?
dybincka [34]
4/15 = 0.267
8/25 = 0.32
0.267/0.32 = 0.834375
3 0
2 years ago
For which angle theta is cos theta =-1?<br>a) 270<br>b) 360<br>c) 450<br>d) 540​
yanalaym [24]

Answer:

540°.

Step-by-step explanation:

On a unit circle, cos θ = -1 at 180°.

However, cos θ has a period of 2π, or 360°. This means that cos θ will equal to -1 again after 2π.

To solve for the angle:

180° + 360° = 540°. This is the next angle at which cos θ = -1.

7 0
3 years ago
Read 2 more answers
Other questions:
  • Suppose h(t)=-5t^2+10t+3 is the height of a diver above the water(in meters), t seconds after the diver leaves the springboard.
    14·1 answer
  • What is the least common multiple 5,6 and 7
    9·2 answers
  • How many 1/2s are in 7 please hurry I’m late for the work
    7·1 answer
  • A village P is 10km from a lorry station,Q on a bearing 065°.another village R is 8km from Q on a bearing 155°.calculate; a.the
    9·1 answer
  • In the diagram above, which two red lines are skew?
    12·1 answer
  • The a value of a function in the form f(x) = ax2 + bx + c is negative. Which statement must be true?
    9·2 answers
  • What present of 25 is 23
    8·1 answer
  • A number is less than twelve
    6·2 answers
  • Add the following weights:<br>6 lb 10 oz<br>+ 10 oz<br>lb<br>OZ​
    8·1 answer
  • Here are some points for all who are working hard and over coming challenges and helping &lt;3 ( p.s there is another one in adv
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!