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RSB [31]
3 years ago
11

Simplify the following expression: 5 + (b + 2) What is the answer?

Mathematics
1 answer:
dolphi86 [110]3 years ago
8 0

Answer:

5 + ( b + 2)

5 + b + 2

5 + 2 + b

7 + b

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Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n.
Vera_Pavlovna [14]

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.

3 0
3 years ago
can someone help me figure this out, There are 16 ounces in one pound. A box of nails weighs 4 pounds. How many ounces does the
Alla [95]
The box weighs 64 ounces
4 0
3 years ago
Read 2 more answers
Can someone solve this for me please.
nika2105 [10]

Answer:

  • The value of PR is 21 ft.

Step-by-step explanation:

<u>We know that:</u>

  • 3(PQ + QT + TP) = PR + RS + SP
  • PQ = 7 ft.
  • QT = 4 ft
  • RS = 12 ft.

<u>Work:</u>

  • 3(PQ + QT + TP) = PR + RS + SP
  • => 3(7 + 4 + TP) = PR + 12 + SP
  • => 21 + 12 + 3TP = PR + 12 + SP
  • => 21 + 3TP = PR + SP
  • => 21 + SP = PR + SP                                                                       [3TP = SP]
  • => 21 = PR

Hence, <u>the value of PR is 21 ft.</u>

Hoped this helped.

BrainiacUser1357

6 0
2 years ago
Choose all the numbers that are multiples of 12.
shusha [124]
Multiples of 12: 12, 36, 48, 60
3 0
3 years ago
9×9 to the power of 18
Nitella [24]
The answer is1.3508517e+18
6 0
3 years ago
Read 2 more answers
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