Question

Find a closed form expression for the nth right Riemann sum of this integral. HELP!?

`int_4^13-4x-5dx`

Answer 1

`-351`

Explanation:

`int_4^13-4x-5dx = -int_4^13(4x+5)dx = -lim_(n->oo)sum_(k=4n)^(k=13n)(4(k/n)+5)1/n = -351`

Answer 2

Please see below.

Explanation:

Step 1
Calm down. There's no need for shouting and exclaiming.

Step 2 All the rest.

For `int_a^b f(x) dx = int_4^13 (-4x-5) dx`

We want `sum_(i=1)^n f(x_i) Delta x`

Find `Delta x`

`Delta x = (b-a)/n = (13-4)/n = 9/n`

Find the right endpoints of the subintervals (`x_i`). For `i = 1, 2, . . . , n`

`x_1 = a+iDeltax = 4+i9/n = 4+(9i)/n`

Find the value of the function (the height, the `y`-value) at each right endpoint.

`f(x_i) = f(4+(9i)/n) = -4(4+(9i)/n)-5 = -21-(36i)/n`

Find the Riemann sum

`sum_(i=1)^n f(x_i) Delta x = sum_(i=1)^n (-21-(36i)/n)(9/n)`

That looks closed form to me, but your grader may want you to simplify this.

`= sum_(i=1)^n (-189/n-(324i)/n^2)`

`= sum_(i=1)^n (-189/n)- sum_(i=1)^n (324i)/n^2)`

`= -189/n sum_(i=1)^n 1 - 324/n^2 sum_(i=1)^n i`

`= -189/n (n) - 324/n^2 ((n(n+1))/2)`

`= -189 - 162 ((n+1)/n)`