$\displaystyle\int_C\mathrm ds = \int_a^b \sqrt{\left(\frac{\mathrm dx}{\mathrm dt}\right)^2+\left(\frac{\mathrm dy}{\mathrm dt}\right)^2} \,\mathrm dt$

In this case, we have

x(t) = exp(t ) + exp(-t ) ==> dx/dt = exp(t ) - exp(-t )

y(t) = 5 - 2t ==> dy/dt = -2

and [a, b] = [0, 2]. The length of the curve is then

$\displaystyle\int_0^2 \sqrt{\left(e^t-e^{-t}\right)^2+(-2)^2} \,\mathrm dt = \int_0^2 \sqrt{e^{2t}-2+e^{-2t}+4}\,\mathrm dt$

$=\displaystyle\int_0^2 \sqrt{e^{2t}+2+e^{-2t}} \,\mathrm dt$

$=\displaystyle\int_0^2\sqrt{\left(e^t+e^{-t}\right)^2} \,\mathrm dt$

$=\displaystyle\int_0^2\left(e^t+e^{-t}\right)\,\mathrm dt$

$=\left(e^t-e^{-t}\right)\bigg|_0^2 = \left(e^2-e^{-2}\right)-\left(e^0-e^{-0}\right) = \boxed{e^2-\frac1{e^2}}$

5 0

2 years ago

Jeff has two sticker-printing machines. Each machine can print 42 stickers per minute. On day 1, only the first machine was used

tankabanditka [31]

All you have to do is x times 42 plus y times 42 and then you have to add the first day on which is 89.This is how it should look (x*42)+(y*42)+89.

3 0

3 years ago

Where v is the final velocity (in m/s), u is the initial velocity (in m/s), a is the acceleration (in m/s²) and s is the distanc

Anna11 [10]

Answer:

The final velocity v is 21.74 m/s.

Step-by-step explanation:

Given:

$\textrm{initial velocity} = u = 9\ m/s\\\textrm{acceleration} = a = 7\ m/s^{2}\\\textrm{distance} = s = 28\ m$

To find:

$\textrm{final velocity} = v = ?$

Solution:

As we know the third equation of motion is represented as

$v^{2} = u^{2} + 2\times a\times s\\$

Substituting the values u, a and s in the above equation we get

$v^{2} = 9^{2} + 2\times 7\times 28\\ =81 + 392\\=473\\$

$v =\±\sqrt{473} \\v =\sqrt{473} \ \textrm{as acceleration is positive v is also positive}\\v = 21.74\ m/s$

7 0

3 years ago

Solve 4x-2y=12 for y Ayo help

Leni [432]

Answer:

y = - 6 + 2x

Step-by-step explanation:

Given

4x - 2y = 12 ( subtract 4x from both sides )

- 2y = 12 - 4x ( divide all terms by - 2 )

y = - 6 + 2x

4 0

2 years ago

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Is (2,10) a solution to this system of inequalities​

Is (2,10) a solution to this system of inequalities