MATHEMATICS COLLEGE

Evaluate the given integral by making an appropriate change of variables, where R is the region in the first quadrant bounded by the ellipse 64x2 + 81y2 = 1. $ L=\iint_{R} {\color{red}9} \sin ({\color{red}384} x^{2} + {\color{red}486} y^{2})\,dA $.

Answers

Answer 1

Answer:

$\displaystyle\iint_R\sin(384x^2+486y^2)\,\mathrm dA$

Notice that Given that $R$ is an ellipse, consider a conversion to polar coordinates:

$\begin{cases}x(r,\theta)=\frac r8\cos\theta\ny(r,\theta)=\frac r9\sin\theta\end{cases}$

The Jacobian for this transformation is

$J=\begin{bmatrix}\frac18\cos\theta&-\frac r8\sin\theta\n\frac19\sin\theta&\frac r9\cos t\end{bmatrix}$

with determinant $\det J=\frac r{72}$

Then the integral in polar coordinates is

$\displaystyle\frac1{72}\int_0^(\pi/2)\int_0^1\sin(6r^2\cos^2t+6r^2\sin^2t)r\,\mathrm dr\,\mathrm d\theta=\int_0^(\pi/2)\int_0^1r\sin(6r^2)\,\mathrm dr\,\mathrm d\theta=\boxed{(\pi\sin^23)/(864)}$

where you can evaluate the remaining integral by substituting $s=6r^2$ and $\mathrm ds=12r\,\mathrm dr$ .

Answer 2

Answer:

Final answer:

To evaluate the integral, we make a change of variables using the transformation x=u/8 and y=v/9 to transform the region into a unit circle. Then we convert the integral to polar coordinates and evaluate it.

Explanation:

To evaluate the given integral, we can make the appropriate change of variables by using the transformation x = u/8 and y = v/9. This will transform the region R into a unit circle. The determinant of the Jacobian of the transformation is 1/72, which we will use to change the differential area element from dA to du dv. Substituting the new variables and limits of integration, the integral becomes:

L = \iint_{R} 9 \sin (612 u^{2} + 768 v^{2}) \cdot (1/72) \,du \,dv

Next, we can convert the integral from Cartesian coordinates(u, v) to polar coordinates (r, \theta). The integral can be rewritten as:

L = \int_{0}^{2\pi} \int_{0}^{1} 9 \sin (612 r^{2} \cos^{2}(\theta) + 768 r^{2} \sin^{2}(\theta)) \cdot (1/72) \cdot r \,dr \,d\theta

We can then evaluate this integral to find the value of L.

Learn more about Evaluation of Integrals here:

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Symbols of:

$\quad\quad\quad\quad\tt{A = A rea}$

$\quad\quad\quad\quad\tt{ l = length}$

$\quad\quad\quad\quad\tt{ b \: = breadth}$

Given that:

$\quad\quad\quad\quad\tt{A = 315 {cm}^(2) }$

$\quad\quad\quad\quad\tt{l = 21cm}$

$\quad\quad\quad\quad\tt{b = \: ? }$

Formula for breadth (b):

$\quad\quad\quad\quad\tt{breadth = (Area)/(length) }$

Solution:

$\quad\quad\quad\quad\tt{b = \frac{315 {cm}^(2) }{21cm} }$

$\quad\quad\quad\tt{\:\:b = {15cm}}$

So, the breadth (b) is:

$\quad\quad\quad\quad\tt \boxed{ \boxed{ \color{magenta}{b = 15cm }}}$

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Answer:

Breadth = 15 cm

Step-by-step explanation:

Area = length x breadth

315 = 21 x breadth

$(315)/(21) = (21)/(21) * breadth$ [ dividing both sides by 21 ]

$15 = 1 * breadth\n\nbreadth = 15 \ cm$

Mario has drawn a plan of his bedroom on 1 cm square paper. His en-suite shower cubicle measure. 1m x1m, give the scale of his drawing as ratio ___ : ___.What are the actual dimension of his bed ____ m x __ m

Answers

Mario has drawn a plan of his bedroom on 1 cm square paper. His en-suite shower cubicle measure. 1m x1m, give the scale of his drawing
as ratio _ 1 cm__ : __1m_.
What are the actual dimension of his bed __1__ m x _1_ m

Answer:

Step-by-step explanation:

1:50

2m:1.5m

Please help please I am begging you

Answers

Answer:

Step-by-step explanation:

9 (x + 1) = 9*x + 9*1 = 9x + 9

A pharmaceutical manufacturer is concerned that the impurity concentration in pills should not exceed 3%. It is known that from a particular production run impurity concentrations follow a normal distribution with a standard deviation of 0.4%. A random sample of 64 pills from a production run was checked, and the sample mean impurity concentration was found to be 3.07%. Testing at the 5% level the null hypothesis that the population mean impurity concentration is 3% against the alternative that it is more than 3%.a. Calculate the p-value for this test.
b. Find the p-value for this test and draw your conclusion.
c. Suppose that the alternative hypothesis in part b had been two-sided rather than one-sided. Conduct your test using either the critical value or the p-value approach.

Answers

The p-value for the given test is 0.0901.

Given that, a pharmaceutical manufacturer is concerned that the impurity concentration in pills should not exceed 3%.

What is a standard deviation?

Standard deviation is the degree of dispersion or the scatter of the data points relative to its mean, in descriptive statistics.

Given, Mean = 3.07, μ=3%, σ =0.4% and n=64.

Hypothesis test $H_(0)$ ;μ=3 and $H_(1)$ ;μ>3

We need to test the null hypothesis against the alternative hypothesis, we reject the null hypothesis and accept the alternative hypothesis.

When x>x critical

x critical = μ+Z+σ

Z = (x-μ)/(σ/√n)

= (3.07-3)/(0.4/√64)

= (0.07)/0.05

= 1.4

α = 5 and $z_{0.05$ = 1.645

If Z> $Z_{0.05$

1.4 > 1.645

Therefore, we do not reject null hypothesis at 5% significant level.

There is sufficient evidence that the mean impurity in concentration pills is 3% or less than 3%.

p-value

p−value=P(Z>1.4)

=1−P(Z<1.4)

=1−0.90988

=0.0901

Therefore, the p-value for the given test is 0.0901.

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Answer:

Step-by-step explanation:

Marl me as brainliest answer c

Describe how to simplify the expression 3^-6/3^-4a. Divide the bases and then add the exponents.
b. Keep the base the same and then add the exponents.
c. Multiply the bases and then subtract the exponents.
d. Keep the base the same and then subtract the exponents.

Answers

Answer:

The correct option is d. To simplify the given expression we should keep the base the same and then subtract the exponents.

Step-by-step explanation:

The given expression is

$(3^(-6))/(3^(-4))$

In the above expression we have common base 3 but the exponents are different.

According to the rule of exponent, if the numerator and denominator have same base and different exponent, then the base remains the same and the exponent of denominator subtracted from exponent of numerator.

$(a^m)/(a^n) =a^(m-n)$