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Stells [14]
3 years ago
13

Which of the following is a mixture?

Physics
1 answer:
rewona [7]3 years ago
5 0
Sand is the mixture because it is a combination of different forms of gravel, stones, and ocean floor sediments.
All other options are chemically bonded, making them solutions or compounds rather than mixtures.
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1.(16 pts.) Find the volume of the solid obtained by revolving the region enclosed by y = xex , y = 0 and x = 1 about the x-axis
MrRa [10]

Answer:

<em>The Volume is 5.018 cubic units</em>

Explanation:

<u>Volume Of A Solid Of Revolution</u>

Let f(x) be a continuous function defined in an interval [a,b], if we take the area enclosed by f(x) between x=a, x=b and revolve it around the x-axis, we get a solid whose volume can be computed as

\displaystyle V=\pi \int_a^bf^2(x)dx

It's called the disk method. There are other available methods to compute the volume.

We have

f(x)=xe^x

And the boundaries defined as x=1, y=0 and revolved around the x-axis. The left endpoint of the integral is easily identified as x=0, because it defines the beginning of the region to revolve. So we need to compute

\displaystyle V=\pi \int_0^1(xe^x)^2dx=\pi \int_0^1x^2e^{2x}dx

We need to first determine the antiderivative

\displaystyle I=\int x^2e^{2x}dx

Let's integrate by parts using the formula

\displaystyle \int u.dv=u.v-\int v.du

We pick u=x^2,\ dv=e^{2x}dx

Then du=2xdx,\ v=\frac{e^{2x}}{2}

Applying by parts:

\displaystyle I=x^2\frac{e^{2x}}{2}-\int 2x\frac{e^{2x}}{2}dx

\displaystyle I=\frac{x^2e^{2x}}{2}-\int xe^{2x}dx

Now we solve

\displaystyle I_1=\int xe^{2x}dx

Making u=x,\ dv=e^{2x}dx

\displaystyle du=dx,\ v=\frac{e^{2x}}{2}

Applying by parts again:

\displaystyle I_1=x\frac{e^{2x}}{2}-\int \frac{e^{2x}}{2}dx

\displaystyle I_1=\frac{xe^{2x}}{2}-\frac{1}{2}\int e^{2x}dx

The last integral is directly computed

\displaystyle \int e^{2x}dx=\frac{e^{2x}}{2}

Replacing every integral computed above

\displaystyle I=\frac{x^2e^{2x}}{2}-\left(\frac{xe^{2x}}{2}-\frac{1}{2}\frac{e^{2x}}{2}\right)

Simplifying

\displaystyle I=\dfrac{\left(2x^2-2x+1\right)\mathrm{e}^{2x}}{4}

Now we compute the definite integral as the volume

V=\pi \left[\dfrac{\left(2(1)^2-2(1)+1\right)\mathrm{e}^{2(1)}-\left(2(0)^2-2(0)+1\right)\mathrm{e}^{2(0)}}{4}\right]

Finally

V=\pi \dfrac{\mathrm{e}^2-1}{4}=5.018

The Volume is 5.018 cubic units

8 0
3 years ago
Which statement about the physical change of liquid water boiling into steam is true
sp2606 [1]
Option 1 would be right one ! as the heat added represent itself into the energy change
3 0
3 years ago
Read 2 more answers
A film of soapy water ( n= 1.33 ) on top of a plastic cutting board has a thickness of 233 nm. What color is most strongly refle
CaHeK987 [17]

Answer:

620nm

Explanation:

4 0
3 years ago
Does the orbital period of a planet depend on the mass of the planet or on the mass of the star that it orbits?
jasenka [17]

Answer:

The orbital period of a planet depends on the mass of the planet.

Explanation:

A less massive planet will take longer to complete one period than a more massive planet.

8 0
2 years ago
Which one of the following accurately describes the force of gravity?
elena55 [62]
Choice ' C ' is a true statement.
The other choices aren't.
8 0
3 years ago
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