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zhenek [66]
2 years ago
13

Sin(x2 y2 da r , where r is the region in the first quadrant between the circles with center the origin and radii 1 and 5

Mathematics
1 answer:
Rudik [331]2 years ago
3 0
I assume there's a plus sign missing above...

Convert to polar coordinates, using

\begin{cases}x(r,\theta)=r\cos\theta\\y(r,\theta)=r\sin\theta\end{cases}

Then the Jacobian is

\dfrac{\partial(x,y)}{\partial(r,\theta)}=\begin{vmatrix}x_r&y_r\\x_\theta&y_\theta\end{vmatrix}=\begin{vmatrix}\cos\theta&\sin\theta\\r\sin\theta&-r\cos\theta\end{vmatrix}=-r

Then

\mathrm dA=\mathrm dx\,\mathrm dy=|-r|\,\mathrm dr\,\mathrm d\theta=r\,\mathrm dr\,\mathrm d\theta

so the integral can be written as

\displaystyle\iint_R\sin(x^2+y^2)\,\mathrm dA=\int_0^{\pi/2}\int_1^5r\sin(r^2)\,\mathrm dr\,\mathrm d\theta

Let s=r^2, so that \dfrac{\mathrm ds}2=r\,\mathrm dr.

\displaystyle\frac12\int_0^{\pi/2}\int_1^{25}\sin s\,\mathrm ds\,\mathrm d\theta=-\frac12(\cos25-\cos1)\int_0^{\pi/2}\mathrm d\theta=\frac\pi4(\cos1-\cos25)
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Step-by-step explanation:

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8 0
2 years ago
A jar of peanut butter contains 454 g with a standard deviation of 10.2 g. Find the probability that a jar contains more than 46
Fofino [41]

Answer:

The probability that a jar contains more than 466 g is 0.119.

Step-by-step explanation:

We are given that a jar of peanut butter contains a mean of 454 g with a standard deviation of 10.2 g.

Let X = <u><em>Amount of peanut butter in a jar</em></u>

The z-score probability distribution for the normal distribution is given by;

                                Z  =  \frac{X-\mu}{\sigma}  ~ N(0,1)

where, \mu = population mean = 454 g

           \sigma = standard deviation = 10.2 g

So, X ~ Normal(\mu=454 , \sigma^{2} = 10.2^{2})

Now, the probability that a jar contains more than 466 g is given by = P(X > 466 g)

            P(X > 466 g) = P( \frac{X-\mu}{\sigma} > \frac{466-454}{10.2} ) = P(Z > 1.18) = 1 - P(Z \leq 1.18)

                                                                  = 1 - 0.881 = <u>0.119</u>

The above probability is calculated by looking at the value of x = 1.18 in the z table which has an area of 0.881.

4 0
2 years ago
Witch of the following could be a horizontal cross-section of a cylinder
vagabundo [1.1K]
The answer is C. Circle

Hope this helps :)
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Step-by-step explanation:

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Which of the following statements describes one part of completing the square for x 2 + 4x = 32?
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SOS:

Answer:

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Step-by-step explanation:

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<em>Hope this helps!!</em>

7 0
2 years ago
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