1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
inna [77]
3 years ago
12

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

Mathematics
1 answer:
Vera_Pavlovna [14]3 years ago
3 0

Split up the integration interval into 4 subintervals:

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

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

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

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

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

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

  • Trapezoidal rule

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

T_i=\dfrac{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)\dfrac{(x-m_i)(x-r_i)}{(\ell_i-m_i)(\ell_i-r_i)}+f(m)\dfrac{(x-\ell_i)(x-r_i)}{(m_i-\ell_i)(m_i-r_i)}+f(r_i)\dfrac{(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.

You might be interested in
John does 5 pushups on the first day of a 30-day month, and then increases the number of pushups by 2 pushups a day. how many pu
xxMikexx [17]
He has done 70 push ups because by the 15th he has done 35 push ups add 15 more days and u get 70 .
7 0
3 years ago
Read 2 more answers
All the dimensions of a pentagon were multiplied by the same factor, and the area of the figure changed by a factor of 25. By wh
elena-s [515]

5


reason is because if you’re pentagon has lines of 5 then the area would be 25
8 0
2 years ago
2/5•35 what is the awnser plz help
diamong [38]
If (.) represent multplication the answer is
2/5×35=14
if (.) represent decimal the answer is
2/ 5.35= 40/107
7 0
4 years ago
A motorcycle rides from point P to Q to R to S and finally to P in a circular path as shown in the below figure. Find a) the dis
Triss [41]

The distance covered by the motorcycle is 31.4 km and displacement will be equal to 0.

<h3>What is Distance and Displacement ?</h3>

Distance and Displacement both are the measure of the movement of an Object from one point to another.

Distance is a scalar quantity while Displacement is a vector quantity , and so it depends upon direction.

Distance travelled will always be positive but Displacement can be positive , negative and zero.

A motorcycle rides from point P to Q to R to S and finally to P in a circular path as shown in the below figure

The missing figure is attached with the answer.

(a) The distance covered by the motorcycle is

2πr = 2 * 3.14 * 5 = 31.4 km

(b) The displacement will be equal to 0 as the starting and the ending point is same.

To know more about Distance and Displacement

https://brainly.in/question/47777661

#SPJ1

6 0
2 years ago
Three of these fractions are equivalent:
Andrews [41]

Answer:

b is the odd one out

Step-by-step explanation:

30/70= .428

12/30= .4

9/21= .428

6/14= .428

all are equal except 12/30

hope this helped!

8 0
2 years ago
Other questions:
  • What is equivalent expression of 6x(5+n)
    5·1 answer
  • What is the value of x in the equation 3x-1/9y=18, when y=27?
    7·1 answer
  • Andy erned $38 on monday and $34 on tuesday. How many lucky bamboo Plants can he buy with the total money he earned?
    10·1 answer
  • Which best describes her prediction?
    14·1 answer
  • A home’s value increases at an average rate of 5.5% each year. The current value is $120,000. What function can be used to find
    14·2 answers
  • A) Antoine's family is taking a summer vacation through the northern united states on day one of the trips the family will be tr
    5·1 answer
  • Helppppppppppppppp will make brainlist
    6·2 answers
  • PLEASE HELP ASAP!
    6·1 answer
  • Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms
    15·1 answer
  • 450 g in the ratio 2:13
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!