Imagine that the light you observe from space is exhibiting a redshift. This would mean that the Universe is

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:

The Volume is 5.018 cubic units

Explanation:

Volume Of A Solid Of Revolution

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

We know that there is a relationship between work and mechanical energy change. Whenever work is done upon an object by an exter

solniwko [45]

Answer:

* roller skates and ice skates.

* roller coaster

Explanation:

One of the best examples for this situation is when we are skating, in the initial part we must create work with a force, it compensates to move, after this the external force stops working and we continue movements with kinetic energy, if there are some ramps, we can going up, where the kinetic energy is transformed into potential energy and when going down again it is transformed into kinetic energy. This is true for both roller skates and ice skates.

Another example is the roller coaster, in this case the motor creates work to increase the energy of the car by raising it, when it reaches the top the motor is disconnected, and all the movement is carried out with changes in kinetic and potential energy. In the upper part the energy is almost all potential, it only has the kinetic energy necessary to continue the movement and in the lower part it is all kinetic; At the end of the tour, the brakes are applied that bring about the non-conservative forces that decrease the mechanical energy, transforming it into heat.

8 0

3 years ago

in your own words provide two advantages of using meters as a measurement of length rather than old measurements of length such

Arlecino [84]

Answer:

First, the meters is the unit used in almost all the world today.

Then using meters as your unit will be easier to read and interpret for a larger number of people than if you use an older system, like hands or steps.

And because of the fact that almost all the world uses the meter as the unit for length, almost all the equations and formulas are written in meters.

Precision: The meter is exactly defined as:

"The length of the path traveled by light in a vacuum in 1/299,792,458 of a second"

So the meter is an exact (and accepted) unit.

Then using this unit for measures of length also provides some " legitimacy" for you measure.

3 0

3 years ago

Whenever you work a physics problem you should get into the habit of thinking about whether the answer is physically realistic.

Marta_Voda [28]

Answer:

Yes, the answer on how far off the ground a typical truck is happens to be physically realistic.

Explanation:

This concept could be explained with when someone is viewing a picture from a particular distance. The farther the person is from the picture, the smaller the details of the picture and the higher they picture would be unappriciated and vice versa.

Likewise the small truck on ground. The distance off ground the truck is being viewed, the smaller the truck would become.

4 0

3 years ago

Read 2 more answers

What do you think we can learn about Earth's history by looking at the Grand Canyon?

viktelen [127]

We can look at all the ages of the earth since it’s a big crack is reveals many layers of the earth and we can know about chemicals and metals that were in earth and diffrent times

5 0

3 years ago