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Ahat [919]
1 year ago
5

For the function given below, find a formula for the Riemann sum obtained by dividing the interval (0, 3) into n equal subinterv

als and us right-hand endpoint for each Then take a limit of this sum as c_{k}; n -> ∞ to calculate the area under the curve over [0, 3] . f(x) = 2x ^ 2 Write a formula for a Riemann sum for the function f(x) = 2x ^ 2 over the interval [0, 3]
Mathematics
1 answer:
Viktor [21]1 year ago
8 0

Splitting up [0, 3] into n equally-spaced subintervals of length \Delta x=\frac{3-0}n = \frac3n gives the partition

\left[0, \dfrac3n\right] \cup \left[\dfrac3n, \dfrac6n\right] \cup \left[\dfrac6n, \dfrac9n\right] \cup \cdots \cup \left[\dfrac{3(n-1)}n, 3\right]

where the right endpoint of the i-th subinterval is given by the sequence

r_i = \dfrac{3i}n

for i\in\{1,2,3,\ldots,n\}.

Then the definite integral is given by the infinite Riemann sum

\displaystyle \int_0^3 2x^2 \, dx = \lim_{n\to\infty} \sum_{i=1}^n 2{r_i}^2 \Delta x \\\\ ~~~~~~~~ = \lim_{n\to\infty} \frac6n \sum_{i=1}^n \left(\frac{3i}n\right)^2 \\\\ ~~~~~~~~ = \lim_{n\to\infty} \frac{54}{n^3} \sum_{i=1}^n i^2 \\\\ ~~~~~~~~ = \lim_{n\to\infty} \frac{54}{n^3}\cdot\frac{n(n+1)(2n+1)}6 = \boxed{18}

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use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine cos^4
Firdavs [7]

The expression cos⁴ θ in terms of the first power of cosine is <u>[ 3 + 2cos 2θ + cos 4θ]/8.</u>

The power-reducing formula, for cosine, is,

cos² θ = (1/2)[1 + cos 2θ].

In the question, we are asked to use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine cos⁴ θ.

We can do it as follows:

cos⁴ θ

= (cos² θ)²

= {(1/2)[1 + cos 2θ]}²

= (1/4)[1 + cos 2θ]²

= (1/4)(1 + 2cos 2θ + cos² 2θ] {Using (a + b)² = a² + 2ab + b²}

= 1/4 + (1/2)cos 2θ + (1/4)(cos ² 2θ)

= 1/4 + (1/2)cos 2θ + (1/4)(1/2)[1 + cos 4θ]

= 1/4 + cos 2θ/4 + 1/8 + cos 4θ/8

= 3/8 + cos 2θ/4 + cos 4θ/8

= [ 3 + 2cos 2θ + cos 4θ]/8.

Thus, the expression cos⁴ θ in terms of the first power of cosine is <u>[ 3 + 2cos 2θ + cos 4θ]/8</u>.

Learn more about reducing trigonometric powers at

brainly.com/question/15202536

#SPJ4

5 0
2 years ago
a local food company produces yogurt in 3/4 tab cups 2 cups= 1 pint 2 pints= 1 quart 4 quarts= 1 gallon 8 fl oz= 1 cup The tubs
miss Akunina [59]

Answer:

15cents.

Step-by-step explanation:

The cost of yogurt is $0.75

20% of this amount is the profit to the company

Profit = \frac{20}{100} * 0.75

          = $0.15

So the profit to the company is 15cents.

7 0
3 years ago
Peter took one of his friends out to lunch. The lunches cost $100 plus he
77julia77 [94]

Answer:

Tax = $12 Tip = $18 Total Cost = $130

Step-by-step explanation:

tax = tax percentage x amount

0.12 x 100 = 12

tip = tip percentage x amount

0.18 x 100 = 18

The Total Cost

total cost = cost + tax + tip

100 + 12 + 18

Add them up and it would be

$130

5 0
3 years ago
Someone help me?Please
Aloiza [94]

Answer:

C

Step-by-step explanation:

7 5/6 is 7.833333333333333333333

5 0
3 years ago
Simplify the expresssion 2x^2+8+3x^2+4x-10
nignag [31]

Answer:

x= 0.348 OR X= -1.148

Step-by-step explanation:

Given that:

2x^2+8+3x^2+4x-10

The first step is to rearrange the parameters given, by doing so, we have:

= 2x^2 +3x^2 +4x -10+8

= 5x^2 +4x - 2

The above equation forms the basis of a quadratic equation;

So using quadratic formula

= (-b +/- (√b^2-4ac))/2a

= (-4 +/- (√4^2 -4*5*(-2))/2*5

= -4 +/- (√ 16 - (-40))/10

=( -4 +/- √ 56 ) / 10

=( -4 +√ 56 ) / 10 OR =( -4 - √ 56 ) / 10

= (-4 + 7.48)/10 OR (-4 -7.48)/10

= 3.48/10 OR -11.48/10

= 0.348 OR. -1.148

SO, x= 0.348 OR X= -1.148

4 0
3 years ago
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