Sue had a circular picture that she wanted to frame. If the picture's frame diameter was

Use the definition to find an expression for the area under the curve y = x3 from 0 to 1 as a limit. lim n→∞ n i = 1 R (b) The f

Harlamova29_29 [7]

The summand (R?) is missing, but we can always come up with another one.

Divide the interval [0, 1] into $n$ subintervals of equal length $\dfrac{1-0}n=\dfrac1n$ :

$[0,1]=\left[0,\dfrac1n\right]\cup\left[\dfrac1n,\dfrac2n\right]\cup\cdots\cup\left[1-\dfrac1n,1\right]$

Let's consider a left-endpoint sum, so that we take values of $f(\ell_i)={\ell_i}^3$ where $\ell_i$ is given by the sequence

$\ell_i=\dfrac{i-1}n$

with $1\le i\le n$ . Then the definite integral is equal to the Riemann sum

$\displaystyle\int_0^1x^3\,\mathrm dx=\lim_{n\to\infty}\sum_{i=1}^n\left(\frac{i-1}n\right)^3\frac{1-0}n$

$=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=1}^n(i-1)^3$

$=\displaystyle\lim_{n\to\infty}\frac1{n^4}\sum_{i=0}^{n-1}i^3$

$=\displaystyle\lim_{n\to\infty}\frac{n^2(n-1)^2}{4n^4}=\boxed{\frac14}$

8 0

3 years ago

What line forms when a plane and a cylinder intersect?

jek_recluse [69]

Hello!!

Parallel lines form when a plane and a cylinder intersects.

Many angles are also produces. And then, the produced angles will together form certain pairs of corresponding angles.

Good luck :)

4 0

3 years ago

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Homework I need help

seraphim [82]

-0.3 is the. correct answer.

8 0

3 years ago

Vince helped serve the hot dogs at the first night’s camp.He served 2 each to 118 peaple and 1 each to 87 people How many dogs d

givi [52]

There are 323 hotdogs total.

7 0

3 years ago

S the following statement always, sometimes, or never true?

liubo4ka [24]

The statement “The supplement of an acute angle is an obtuse angle.” is always true because the sum of any two supplementary angles is 180°, so if one of them less than 90° (acute), then the other must be greater than 90° (obtuse)

Step-by-step explanation:

Let us revise some facts about the supplementary angles

Supplementary angles are two angles the sum of their measure is 180°
The two angles could be right angles or one of them is acute and the other is obtuse
The two angles can not be acute angles or obtuse angles

Let us take some examples to explain the facts above

∵ One of two supplementary angles is a right angle

∵ The measure of the right angle is 90°

∵ The sum of the measures of the supplementary angles is 180°

∴ 90 + the measure of the supplement angle = 180

- Subtract 90 from both sides

∴ The measure of the supplement angle = 90°

∴ The supplement of a right angle must be a right angle

∵ One of two supplementary angles is 70°

∵ Its measure less than 90°

∴ This angle is an acute angle

∵ The sum of the measures of the supplementary angles is 180°

∴ 70 + the measure of the supplement angle = 180

- Subtract 70 from both sides

∴ The measure of the supplement angle = 110°

∵ Its measure greater than 90°

∴ The supplement angle is an obtuse angle

∴ The supplement of an acute angle must be an obtuse angle

The two supplementary angles can not be acute angles because their sum is less than 180°

The two supplementary angles can not be obtuse angles because their sum is greater than 180°

The statement “The supplement of an acute angle is an obtuse angle.” is always true because the sum of any two supplementary angles is 180°, so if one of them less than 90° (acute), then the other must be greater than 90° (obtuse)

Learn more:

You can learn more about the supplementary angles in brainly.com/question/11175936

#LearnwithBrainly

4 0

3 years ago

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