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balu736 [363]
2 years ago
13

Question 2b only! Evaluate using the definition of the definite integral(that means using the limit of a Riemann sum

Mathematics
1 answer:
lara [203]2 years ago
4 0

Answer:

Hello,

Step-by-step explanation:

We divide the interval [a b] in n equal parts.

\Delta x=\dfrac{b-a}{n} \\\\x_i=a+\Delta x *i \ for\ i=1\ to\ n\\\\y_i=x_i^2=(a+\Delta x *i)^2=a^2+(\Delta x *i)^2+2*a*\Delta x *i\\\\\\Area\ of\ i^{th} \ rectangle=R(x_i)=\Delta x * y_i\\

\displaystyle \sum_{i=1}^{n} R(x_i)=\dfrac{b-a}{n}*\sum_{i=1}^{n}\  (a^2 +(\dfrac{b-a}{n})^2*i^2+2*a*\dfrac{b-a}{n}*i)\\

=(b-a)^2*a^2+(\dfrac{b-a}{n})^3*\dfrac{n(n+1)(2n+1)}{6} +2*a*(\dfrac{b-a}{n})^2*\dfrac{n (n+1)} {2} \\\\\displaystyle \int\limits^a_b {x^2} \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} R(x_i)\\\\=(b-a)*a^2+\dfrac{(b-a)^3 }{3} +a(b-a)^2\\\\=a^2b-a^3+\dfrac{1}{3} (b^3-3ab^2+3a^2b-a^3)+a^3+ab^2-2a^2b\\\\=\dfrac{b^3}{3}-ab^2+ab^2+a^2b+a^2b-2a^2b-\dfrac{a^3}{3}  \\\\\\\boxed{\int\limits^a_b {x^2} \, dx =\dfrac{b^3}{3} -\dfrac{a^3}{3}}\\

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Suppose that the distribution of typing speed in words per minute (wpm) for experienced typists using a new type of split keyboa
KengaRu [80]

Answer:

a. <u>0.5 or 50%</u>

b. <u>0.496 or 49.6%</u>

c. <u>0.8185 or 81.85%</u>

d. <u>Yes, it would be just 0.0013 or 0.13% of probability to find a typist whose speed exceeds 105 wpm</u>

e. <u>0.0252 or 2.52%</u>

f. <u>The qualifying speed would be 47.4 wpm.</u>

Step-by-step explanation:

a. Let's find the z-score this way:

μ  = 60 σ= 15

z-score = (x - μ)/σ

z-score = (60 - 60)/15

z-score = 0

Now, let's calculate the p value for z-score = 0, using the z-table:

<u>p (z=0) = 0.5 or 50%</u>

b. z-score = (x - μ)/σ

z-score = (59.9 - 60)/15

z-score = -0.01

Now, let's calculate the p value for z-score = -.0.01, using the z-table:

<u>p (z = -0.01) = 0.496 or 49.6%</u>

If the question were less or equal than 60, a and b would have the same answer. But in this case, the question is "less than 60 wpm".

c.

z-score = (x - μ)/σ

z-score = (45 - 60)/15

z-score = - 1

Now, let's calculate the p value for z-score = -1, using the z-table:

p (z = -1) = 0.1587

z-score = (x - μ)/σ

z-score = (90 - 60)/15

z-score = 2

Now, let's calculate the p value for z-score = 2, using the z-table:

p (z = 2) = 0.9772

<u>In consequence,</u>

<u>p (-1 ≤ z ≤ 2) = 0.9772 - 0.1587 = 0.8185 or 81.85%</u>

d. z-score = (x - μ)/σ

z-score = (105 - 60)/15

z-score = 3

Now, let's calculate the p value for z-score = 3, using the z-table:

p (z = 3) = 0.9987

In consequence,

p (z > 3) = 1 - 0.9987 = 0.0013

<u>Yes, it would be just a 0.13% of probability to find a typist whose speed exceeds 105 wpm.</u>

e. z-score = (x - μ)/σ

z-score = (75 - 60)/15

z-score = 1

Now, let's calculate the p value for z-score = 1, using the z-table:

p (z = 1) = 0.8413

In consequence,

p (z > 1) = 1 - 0.8413 = 0.1587

and if the two typists are independently selected, then

p = 0.1587 * 0.1587

<u>p = 0.0252 or 2.52%</u>

f. p = 0.2, using the z-table, the z-score is -0.84, then:

z-score = (x - μ)/σ

-0.84 = (x - 60)/15

-12.6 = x - 60

-x = -60 + 12.6

-x = - 47.4

x = 47.4

<u>The qualifying speed would be 47.4 wpm.</u>

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Mason and his dad are going to build a small retaining wall for the perimeter of their new
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Answer:

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Answer:

5

Step-by-step explanation:

a^2 + b^2 = c^2

4^2 + 3^2 = c^2

16 + 9 = c^2

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Answer:

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Step-by-step explanation:

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