Calculus Question

Question

Consider the definite integral ∫¹³  (-4x-5)dx  Find a closed form expression for the nth right Riemann sum of this integral.           

                                            ⁴                            HELP!?

Answer 1

Δx = (13-4)/n = 9/n   f(x) = -4x - 5 

 

right endpoint of first interval  = x₁ = 4 + 9/n

right endpoint of 2nd interval = x₂ = 4 + 2(9/n)

...

right endpoint of interval i = x_i = 4 + i(9/n) = 4 + (9i)/n

 

Right Riemann Sum = ∑_{(i=1 to n)} f(x_i)Δx

 

                              = ∑_{(i=1 to n)} [-4-5x_i](9/n)

 

                              = (9/n)∑_{(i=1 to n)} [-4 - 5(4+(9i)/n)]

 

                              = (9/n)∑_{(i=1 to n)} [-24 - 45i/n]

 

Answer 2

Lets determine how rectangles we should use.  The number of rectangles must be a whole number.

 

base length = (13 - 4) / N

 1 = 9 / N

 

N = 9

 

We will use 9 rectangles with base length of 1.  Since the slope is negative and the curve is below the x-axis, we will use left corner rectangular areas.

 

1st area = f(4)

2nd area = f(5)

3rd area = f(6)

4th area = f(7)

5th area = f(8)

6th area = f(9)
7th area = f(10)

8th area = f(11)

9th area = f(12)

 

Add up these areas to get the area under the curve.