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
What are the variables in the expression below?
N76 [4]

Answer:

pizza

Step-by-step explanation:

dough sauce cheese toppings ∞║║║║║║║║

3 0
3 years ago
165 is what percent of 750?​
Elza [17]

Your answer is...…

(ノ◕ヮ◕)ノ*:・゚✧ ✧゚・: *ヽ(◕ヮ◕ヽ)

22!

3 0
3 years ago
Read 2 more answers
A store owner decided to collect data on his customers. On one day he observed:
Mnenie [13.5K]

Answer:

Johannes Gutenberg - Wikipediahttps://en.wikipedia.org › wiki › Johannes_Gutenberg

Johannes Gensfleisch zur Laden zum Gutenberg was a German goldsmith, inventor, printer, and publisher who introduced printing to Europe with his ...

Occupation: Engraver, ‎inventor‎, and printer‎

Born: Johannes Gensfleisch zur Laden zum G...

Died: February 3, 1468 (aged about 68); Mainz‎, ...‎

Known for: The invention of the movable-type ...

‎Early life · ‎Printing method with... · ‎Legacy · ‎References

People also ask

Step-by-step explanation:

6 0
2 years ago
What is the y-intercept of the line shown below? <br> A. 5<br> B. -5<br> C. 2<br> D. -2
lys-0071 [83]

Answer: -2

Step-by-step explanation: The y-intercept of a line is the point where the graph crosses or intersects the y-axis.

In this case, the line crosses the y-axis 2 units down from the

origin and we can write the coordinates of the point as (0, -2).

So the answer is -2.

3 0
2 years ago
Read 2 more answers
How to solve 3.5x = 14.7​
Anna [14]

Answer:

×=4.2

Step-by-step explanation:

3.5x=14.7

÷3.5 ÷3.5

x=4.2

6 0
3 years ago
Read 2 more answers
Other questions:
  • Which is the best approximation for the solution of the system of equations? y = x + 1 y = 3x – 2
    13·2 answers
  • A researcher wishes to determine the average number of vehicles are registered to a typical Houston residence. In order to do th
    12·1 answer
  • 6.soruyu çözer misiniz lütfen? Bu arada sağdaki okun sonunda E yazıyor. 65 PUAN!!!
    11·2 answers
  • How would I do this confused
    5·2 answers
  • Can I have some topical past paper questions on the olevel syllabus of cambridge? The code is 4024.​
    7·1 answer
  • Identify a pattern by making a table of the inputs and outputs.
    12·1 answer
  • 1. What is the volume of 0.75 liters expressed in mm cubed?​
    14·1 answer
  • HELP ME PLEASE! I will give brainliest!!!
    7·1 answer
  • APML - ATAQ. Solve for x and QR<br><br> HELP ASAP!!!!! PLZZZZZ
    11·1 answer
  • PLEASE HELP MEEEEEE PLEASEEEEEEEE!!!!!! IM TIMED
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!