Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) π/2 0 3 1 + cos(x) dx, n = 4

Answers

Answer 1

Answer:

Split up the integration interval into 4 subintervals:

$\left[0,\frac\pi8\right],\left[\frac\pi8,\frac\pi4\right],\left[\frac\pi4,\frac{3\pi}8\right],\left[\frac{3\pi}8,\frac\pi2\right]$

The left and right endpoints of the $i$ -th subinterval, respectively, are

$\ell_i=\frac{i-1}4\left(\frac\pi2-0\right)=\frac{(i-1)\pi}8$

$r_i=\frac i4\left(\frac\pi2-0\right)=\frac{i\pi}8$

for $1\le i\le4$ , and the respective midpoints are

$m_i=\frac{\ell_i+r_i}2=\frac{(2i-1)\pi}8$

Trapezoidal rule

We approximate the (signed) area under the curve over each subinterval by

$T_i=\frac{f(\ell_i)+f(r_i)}2(\ell_i-r_i)$

so that

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4T_i\approx\boxed{3.038078}$

Midpoint rule

We approximate the area for each subinterval by

$M_i=f(m_i)(\ell_i-r_i)$

so that

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4M_i\approx\boxed{2.981137}$

Simpson's rule

We first interpolate the integrand over each subinterval by a quadratic polynomial $p_i(x)$ , where

$p_i(x)=f(\ell_i)((x-m_i)(x-r_i))/((\ell_i-m_i)(\ell_i-r_i))+f(m)((x-\ell_i)(x-r_i))/((m_i-\ell_i)(m_i-r_i))+f(r_i)((x-\ell_i)(x-m_i))/((r_i-\ell_i)(r_i-m_i))$

so that

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4\int_(\ell_i)^(r_i)p_i(x)\,\mathrm dx$

It so happens that the integral of $p_i(x)$ reduces nicely to the form you're probably more familiar with,

$S_i=\displaystyle\int_(\ell_i)^(r_i)p_i(x)\,\mathrm dx=\frac{r_i-\ell_i}6(f(\ell_i)+4f(m_i)+f(r_i))$

Then the integral is approximately

$\displaystyle\int_0^(\pi/2)\frac3{1+\cos x}\,\mathrm dx\approx\sum_(i=1)^4S_i\approx\boxed{3.000117}$

Compare these to the actual value of the integral, 3. I've included plots of the approximations below.

Answer 2

Answer:

Final answer:

The question is asking to approximate the definite integral of 1 + cos(x) from 0 to π/2 using the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule for n=4. These are numerical methods used for approximating integrals by estimating the area under the curve as simpler shapes.

Explanation:

This question asks to use several mathematical rules, specifically the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule, to approximate the given integral with a specified value of n which is 4. The integral given is the function 1 + cos(x) dx from 0 to π/2. Each of these rules are techniques for approximating the definite integral of a function. They work by estimating the region under the graph of the function and above the x-axis as a series of simpler shapes, such as trapezoids or parabolas, and then calculating the area of these shapes. The 'dx' component represents a small change in x, the variable of integration. The cosine function in this integral is a trigonometric function that oscillates between -1 and 1, mapping the unit circle to the x-axis. The exact solution would require calculus, but these numerical methods provide a close approximation.

Learn more about Numerical Integration Rules here:

brainly.com/question/36635050

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en una canasta se tienen 10 bolas cafes, 5 bolas azules y 15 verdes. Si se saca una al azar, ¿cual es la probabilidad de que esta no sea azul? ¿cual es probabilidad de que sea verde?

Answers

Answer:

The probability that the selected ball is not blue is $(5)/(6)$ .

The probability that the selected ball is green is $(1)/(2)$ .

Step-by-step explanation:

The question is:

There are 10 brown balls, 5 blue balls and 15 green balls in a basket. If one is drawn at random, what is the probability that it is not blue? What is the probability that it is green?

Solution:

The probability of an event E is the ratio of the favorable number of outcomes to the total number of outcomes.

$P(E)=(n(E))/(N)$

The probability of the given event not taking place is known as the complement of that event.

Complement of the event E is,

1 – P (E)

The number of different color balls are as follows:

Brown = n (Br) = 10

Blue = n (Bu) = 5

Green = n (G) = 15

Total = N = 30

Compute the probability of selecting a blue ball as follows:

$P(\text{Bu})=\frac{n(\text{Bu})}{N}=(5)/(30)=(1)/(6)$

Compute the probability of not selecting a blue ball as follows:

$P(\text{Not Bu})=1-P(\text{Bu})$

$=1-(1)/(6)\n\n=(6-1)/(6)\n\n=(5)/(6)$

Thus, the probability that the selected ball is not blue is $(5)/(6)$ .

Compute the probability of selecting a green ball as follows:

$P(\text{G})=\frac{n(\text{G})}{N}=(15)/(30)=(1)/(2)$

Thus, the probability that the selected ball is green is $(1)/(2)$ .

234 base five
Writen in base 10

Answers

$234_5=2*5^2+3*5^1+4*5^0=50+15+4=69_(10)$

Find the midpoint of the segment with the endpoints: (-2, 6) and (-3, 7)

Answers

Answer:

The midpoint is ( -2.5, 6.5)

Step-by-step explanation:

To find the x coordinate of the midpoint, add the x coordinates of the endpoints and divide by 2

(-2+-3)/2 = -5/2 = -2.5

To find the y coordinate of the midpoint, add the y coordinates of the endpoints and divide by 2

(6+7)/2 = 13/2 = 6.5

The midpoint is ( -2.5, 6.5)

Two times Noura's age plus 5 years equals Ali's age. Ali is 19 years old. What formula could be used to calculate Noura's age

Answers

Answer:

Noura is seven years old

Step-by-step explanation:

to get Noura's age first you need to subtract 5 years from Ali's age which will get you to 14 years of age now you just need to divide 14 by two to get Noura's age which is seven

I need help on this question please and thank you that would be great

Answers

Answer:

w independant a dependant

Step-by-step explanation:

A cell site is a site where electronic communications equipment is placed in a cellular network for the use of mobile phones. The numbers y of cell sites from 1985 through 2011 can be modeled byy = 269573/1+985e^-0.308t where t represents the year, with t = 5 corresponding to 1985. Use the model to find the numbers of cell sites in the years 1998, 2003, and 2006.