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
Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters and the other is
JulijaS [17]
A)=14 meters because 3.5 times 4 is 14
b)=4 meters because circles don't have any sides
c)=a would give her more space to plant because 14 is bigger than 4
4 0
3 years ago
Read 2 more answers
SHOW YOUR WORK and ill give brainlest :)
Rashid [163]

The price of the Xbox One after tax is $265

<h3><u>Solution:</u></h3>

The price of a Xbox one is $250

The tax on Xbox one is 6%  

Now, the tax of 6% on 250 is calculated as follows:-

=\frac{6}{100} \times 250\\\\=\frac{250 \times 6}{100}=\frac{1500}{100}=15

So, the tax on Xbox one is $15

The total price of box including tax can be calculated by adding price of box and tax on box

<em>Total price of Xbox after tax = actual price of the Xbox One  + Tax on the Xbox One </em>

= $250 + $15

= $265

Hence, the price of the Xbox One after tax is $265

3 0
3 years ago
Carly is ordering candles online.
kirza4 [7]

Answer:

Is that the whole question or....

7 0
3 years ago
Simplify square root of five times the quantity six minus four square root of three.
NemiM [27]

6√5 - 4√3-

i took the test but sorry if its wrong

5 0
3 years ago
A package states that there are 60 calories in 12 crackers and 75 calories in 15 crackers. Since the relationship is proportiona
AveGali [126]

Answer:

900 calories

Step-by-step explanation:

First you have to find the unit rate, how many calories are there in ONE cracker? To find that you have to divide 60 ÷ 12 = 5 and to check with the other one, 75 ÷ 15 = 5

So, one cracker is 5 calories. Now, how many calories are in 180 crackers? You multiply, 180 x 5 = 900

That means there are 900 calories in 180 crackers.

Hope that helps and have a great day!

5 0
3 years ago
Other questions:
  • Is this a nonlinear function xy=2x-1
    10·1 answer
  • Please help me. Please!!
    5·2 answers
  • What is the value of y in the triangle?
    7·1 answer
  • How do you graph and inequality?​
    10·1 answer
  • What is the probability that the system does not have a type 2defect?b.What is the probability that the system has both type 1 a
    6·1 answer
  • Jamel wants to buy a PS 5. He found it for $500
    9·1 answer
  • ???? Help me and I’ll mark as brainliest
    15·2 answers
  • Translation Guide: 5=A - 1 = L - 4 = Y - 5+8 3(-2)-3 -6=1 2=U 8+6 4-10 - 2 = H 15 = T 3(-2) 2(6) + 2 -9=S 20-18 12 = F 7 = R 3=
    10·1 answer
  • Name 3 pairs of congruent angles.
    6·1 answer
  • wo competing companies are having sales on projectors. Projector #MD375 has a retail price of $1,200. The Office Spot is adverti
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!