Question

CALCULUS HELP ASAP

Write the limit as n goes to infinity of the summation from k equals 1 of the product of the 10th power of the quantity 5 plus 2 times k over n and 2 over n as a definite integral.


THANK YOU!

Answer 1

Answer: (a = 5, b = 7)∫(x^10)dx

Explanation:

I've attached an image of my notes when I learned this. The (5+k2/n) represents x. Since there is a ^10 outside of (5+k2/n), the function is simply x^10. To find the bounds, we can look at the formula (a + k(b-a)/n).

a is easy to find since you don't need to solve for anything. a = 5. To find b, look at the numerator of the fraction.

b - a = 2

b - 5 = 2

b = 7

Answer 2

`\displaystyle \int^7_5 x^(10)\; \text{d}x`

Explanation:

The Riemann sum is a method by which we can approximate the area under a curve using a series of rectangles.

Definite Integral Notation (Reimann Sum)

The area under the curve of f(x) on the interval [a, b] is represented by:

`\boxed{\begin{minipage}{6cm}$\displaystyle \int^b_af(x)\; \text{d}x=\lim_(n \to \infty)\sum^n_(i=1)f(x_i) \cdot \Delta x$\\\\\\where $\Delta x=(b-a)/(n)$ and $x_i=a+\Delta x \cdot i\\\end{minipage}}`

Therefore:

`\begin{aligned}\displaystyle \int^b_af(x)\; \text{d}x&=\lim_(n \to \infty)\sum^n_(i=1)f(x_i) \cdot \Delta x\\\\&=\lim_(n \to \infty)\sum^n_(i=1)f\left(a+\left((b-a)/(n)\right)i\right) \cdot \left((b-a)/(n)\right)\end{aligned}`

Given Reimann sum:

`\displaystyle \lim_(n \to \infty)\sum^n_(k=1)\left(5+k(2)/(n)\right)^(10) (2)/(n)`

For the given Reimann sum:

`\displaystyle \lim_(n \to \infty)\sum^n_(i=1)f\left(a+\left((b-a)/(n)\right)i\right) \cdot \left((b-a)/(n)\right)=\lim_(n \to \infty)\sum^n_(k=1)\left(5+k(2)/(n)\right)^(10) (2)/(n)`

Therefore:

`\Delta x=(b-a)/(n)=(2)/(n)`

`k\textsf{th\;term}=\left(5+k(2)/(n)\right)^(10)`

`a=5`

This means that f(x) is:

`f(x_k)=f(a+k \Delta x)=f\left(5+k(2)/(n)\right)\implies \boxed{f(x)=x^(10)}`

Use the value of a to find the value of b:

`b-a=2`

`b-5=2`

`b=7`

Therefore, the limits for the definite integral are a = 5 and b = 7.

Substitute the found values into the definite integral formula to rewrite the given Reimann sum as a definite integral:

`\displaystyle \lim_(n \to \infty)\sum^n_(k=1)\left(5+k(2)/(n)\right)^(10) (2)/(n)=\boxed{\int^7_5 x^(10)\; \text{d}x}`