18. Xander's water bottle has 15.4 oz of water

The expression of integral as a limit of Riemann sums of given integral $\int\limits^5_b {1} \, x/(2+x^{3}) dx$ is 4 $\lim_{n \to \infty}$ ∑ $n(n+4i)/2n^{3}+(n+4i)^{3}$ from i=1 to i=n.

Given an integral $\int\limits^5_b {1} \, x/(2+x^{3}) dx$ .

We are required to express the integral as a limit of Riemann sums.

An integral basically assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinite data.

A Riemann sum is basically a certain kind of approximation of an integral by a finite sum.

Using Riemann sums, we have :

$\int\limits^b_a {f(x)} \, dx$ = $\lim_{n \to \infty}$ ∑f(a+iΔx)Δx ,here Δx=(b-a)/n

$\int\limits^5_1 {x/(2+x^{3}) } \, dx$ =f(x)= $x/2+x^{3}$

⇒Δx=(5-1)/n=4/n

f(a+iΔx)=f(1+4i/n)

f(1+4i/n)= $[n^{2}(n+4i)]/2n^{3}+(n+4i)^{3}$

$\lim_{n \to \infty}$ ∑f(a+iΔx)Δx=

$\lim_{n \to \infty}$ ∑ $n^{2}(n+4i)/2n^{3}+(n+4i)^{3}4/n$

=4 $\lim_{n \to \infty}$ ∑ $n(n+4i)/2n^{3}+(n+4i)^{3}$

Hence the expression of integral as a limit of Riemann sums of given integral $\int\limits^5_b {1} \, x/(2+x^{3}) dx$ is 4 $\lim_{n \to \infty}$ ∑ $n(n+4i)/2n^{3}+(n+4i)^{3}$ from i=1 to i=n.

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5 0

1 year ago

Which point on the number line represents the approximate volume of a cone with the radius of 3 units and a height of 4 units? U

alekssr [168]

Answer: Third red dot after 30 on the number line represents the approximate volume of a cone.

Step-by-step explanation:

Since we have have given that

Radius of cone = 3 units

Height of cone = 4 units

So, Volume of cone is given by

$Volume=\frac{1}{3}\pi^2\times h\\\\Volume=\frac{1}{3}\times 3.14\times 3\times 3\times 4\\\\Volume=37.68\ units^3$

Hence, Third red dot after 30 on the number line represents the approximate volume of a cone.

5 0

2 years ago

Read 2 more answers

A teacher asked the class to solve the equation 3(times+2)= 21 Robert wrote 3 times + 6=21 as his first step. Which property did

suter [353]

He used 3 distributive property :)

3 0

2 years ago