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romanna [79]
3 years ago
9

The region in the first quadrant bounded by the x-axis, the line x = ln(π), and the curve y = sin(e^x) is rotated about the x-ax

is. What is the volume of the generated solid?
Mathematics
1 answer:
charle [14.2K]3 years ago
5 0
First, it would be good to know that the area bounded by the curve and the x-axis is convergent to begin with.

\displaystyle\int_{-\infty}^{\ln\pi}\sin(e^x)\,\mathrm dx

Let u=e^x, so that \mathrm dx=\dfrac{\mathrm du}u, and the integral is equivalent to

\displaystyle\int_{u=0}^{u=\pi}\frac{\sin u}u\,\mathrm du

The integrand is continuous everywhere except u=0, but that's okay because we have \lim\limits_{u\to0^+}\frac{\sin u}u=1. This means the integral is convergent - great! (Moreover, there's a special function designed to handle this sort of integral, aptly named the "sine integral function".)

Now, to compute the volume. Via the disk method, we have a volume given by the integral

\displaystyle\pi\int_{-\infty}^{\ln\pi}\sin^2(e^x)\,\mathrm dx

By the same substitution as before, we can write this as

\displaystyle\pi\int_0^\pi\frac{\sin^2u}u\,\mathrm du

The half-angle identity for sine allows us to rewrite as

\displaystyle\pi\int_0^\pi\frac{1-\cos2u}{2u}\,\mathrm du

and replacing v=2u, \dfrac{\mathrm dv}2=\mathrm du, we have

\displaystyle\frac\pi2\int_0^{2\pi}\frac{1-\cos v}v\,\mathrm dv

Like the previous, this require a special function in order to express it in a closed form. You would find that its value is

\dfrac\pi2(\gamma-\mbox{Ci}(2\pi)+\ln(2\pi))

where \gamma is the Euler-Mascheroni constant and \mbox{Ci} denotes the cosine integral function.
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The equation of a line is -6x - 2y = -18. What is the x-intercept & y-intercept of the line?
WARRIOR [948]

Answer: y int: (0,9)    x int: (3,0)

Step-by-step explanation:

In slope intercept form, the equation is y=-3x+9. In the formula y=mx+b, we know b is the y intercept, so our y int. is 9. To find our x intercept, we set y=0. So, 0=-3x+9=>3x=9=>x=3

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1. Which of the following measurements could be the three side lengths of a right triangle? a. 4 cm, 5 cm, 9 cm b. 12 cm, 20 cm,
HACTEHA [7]

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b and c

Step-by-step explanation:

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In B, 12+20 is 32, which is greater than 25. And in C, 18+24 is 42, which is greater than 30, so they both will form a triangle.

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The present age of the father is twice the age of his daughter. Ten years ago, the
Elenna [48]

Answer:

<h3>           The father is 40 and the daughter is 20.</h3>

Step-by-step explanation:

x           -  the present age of the daughter

2x         -  the present age of the fathter  

x - 10          -  the age of the daughter ten year ago

2x - 10        -  the age of the fathter ten year ago

Father is older than his dauther, so:

2x - 10 = (x - 10) + 20  

2x - 10 = x - 10 + 20  

2x - 10 = x + 10              {subtract x from both sides}

x - 10 = 10                      {add 10 to both sides}

x = 20

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6 0
3 years ago
Make u the subject v squared= u squared 2as
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Plz make ur question a little bit more understandable, but what I understand is 

u(v²) = u²

u = u/v

then if you need to simplify it then

u = 1/v

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