John is having a barbeque and purchased 12.25 pounds of hamburger for $3.12 per pound. How much did John spend on

Consider the following theorem. Theorem If f is integrable on [a, b], then b a f(x) dx = lim n→[infinity] n i = 1 f(xi)Δx where

mel-nik [20]

Split up the interval [1, 9] into n subintervals of equal length (9 - 1)/n = 8/n :

[1, 1 + 8/n], [1 + 8/n, 1 + 16/n], [1 + 16/n, 1 + 24/n], …, [1 + 8 (n - 1)/n, 9]

It should be clear that the left endpoint of each subinterval make up an arithmetic sequence, so that the i-th subinterval has left endpoint

1 + 8/n (i - 1)

Then we approximate the definite integral by the sum of the areas of n rectangles with length 8/n and height $f(x_i)$ :

$\displaystyle \int_1^9 (x^2-4x+6) \,\mathrm dx \approx \sum_{i=1}^n \frac8n\left(\left(1+\frac8n(i-1)\right)^2-4\left(1+\frac8n(i-1)\right)+6\right)$

Take the limit as n approaches infinity and the approximation becomes exact. So we have

$\displaystyle \int_1^9 (x^2-4x+6) \,\mathrm dx = \lim_{n\to\infty} \sum_{i=1}^n \frac8n\left(\left(1+\frac8n(i-1)\right)^2-4\left(1+\frac8n(i-1)\right)+6\right) \\\\ = \lim_{n\to\infty} \frac8n \sum_{i=1}^n \left(1+\frac{16}n(i-1)+\frac{64}{n^2}(i-1)^2-4-\frac{32}n(i-1)+6\right) \\\\= \lim_{n\to\infty} \frac8{n^3} \sum_{i=1}^n \left(64(i-1)^2-16n(i-1)+3n^2\right) \\\\= \lim_{n\to\infty} \frac8{n^3} \sum_{i=0}^{n-1} \left(64i^2-16ni+3n^2\right) \\\\= \lim_{n\to\infty} \frac8{n^3} \left(64\sum_{i=0}^{n-1}i^2 - 16n\sum_{i=0}^{n-1}i + 3n^2\sum{i=0}^{n-1}1\right) \\\\= \lim_{n\to\infty} \frac8{n^3} \left(\frac{64(2n-1)n(n-1)}{6} - \frac{16n^2(n-1)}{2} + 3n^3\right) \\\\= \lim_{n\to\infty} \frac8{n^3} \left(\frac{49n^3}3-24n^2+\frac{32n}3\right) \\\\= \lim_{n\to\infty} \frac{8\left(49n^2-72n+32\right)}{3n^2} = \boxed{\frac{392}3}$

3 0

3 years ago

Calculating the relative frequencies from the data given in the table. choose all that correctly describe an association between

adoni [48]

Answer:

a higher percentage of 8th graders than 7th graders prefer history.

a higher percentage of 8th graders than 7th graders prefer math/science.

Step-by-step explanation:

6 0

2 years ago

Of the four choices given, which two, when written as a system, have a solution of (–4, 5)? the options and A), B), C), and D) a

Oksi-84 [34.3K]

Yes it sure is A! Even though I’m not answering the question I’m sorry but I’m going by what the person before me just said so I believe them! Hope this helps!

7 0

2 years ago

What is the value of 6.7x 0.54=

xxMikexx [17]

Answer:

The answer is 3.618

Step-by-step explanation:

8 0

3 years ago

Read 2 more answers

The masses of the oranges on sale at a farm stand are normally distributed with a mean of 239 grams and a standard deviation of

ra1l [238]

Answer:

0.8

Step-by-step explanation:

i just took the test

8 0

3 years ago