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

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

3 years ago

408 rounded to the nearest tenth

andreev551 [17]

408 round to the nearest tenth: 410

3 0

3 years ago

Agata [3.3K]

Answers:

prime number = {2,3,5}
composite number = {4,6}
number less than four = {1,2,3}
number more than or equal to three = {3,4,5,6}

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Explanation:

The list of all possible outcomes on a single standard die is {1,2,3,4,5,6}. The curly braces indicate set notation.

A prime number is a number where its only factors are 1 and itself. The value 1 is not prime, and it's not composite either.

Something like 3 is prime because its only factors are 1 and 3. Something like 4 is composite because 4 = 2*2. A composite value has factors other than 1 and itself.

Parts (iii) and (iv) are fairly straight forward. You simply list items less than four for part (iii) and you list items that are three or greater for part (iv).

5 0

3 years ago