How do you calculate the right hand and left hand riemann sum using 4 sub intervals of #f(x)= 3x# on the interval [1,5]?

Question

Charles Arocha · Accepted Answer

See the explanation section, below.

#f(x) = 3x# #[a,b] = [1,5]# and #n=4# Assuming that we are using subintervals of equal length, we get: #Deltax = (b-a)/n = (5-1)/4 = 1# Endpoints of the subintervals are found by starting at #a# and successively adding #Delta x# until we reach #b# The endpoints are #1,2,3,4,5# (The subintervals are: #[1,2], [2,3], [3,4], [4,5]# The left endpoints are #1,2,3,4# #L_4 = f(1)Deltax + f(2)Deltax + f(3)Deltax + f(4)Deltax # (Do the arithmetic.) The right endpoints are #2, 3, 4, 5# #R_4 = f(2)Deltax + f(3)Deltax + f(4)Deltax + f(5)Deltax# (Do the arithmetic.)

Charles Arocha · Answer

undefined To calculate the Right-Hand and Left-Hand Riemann sums for $f(x) = 3x$ on the interval $[1,5]$ with 4 subintervals, follow these steps:

### Step 1: Calculate the width of each subinterval

- The total width of the interval from 1 to 5 is $5 - 1 = 4$.
- With 4 subintervals, each has a width ($\Delta x$) of $\frac{4}{4} = 1$.

### Step 2: Identify the endpoints for each subinterval

- **Subintervals**:
  - 1st: $[1,2]$
  - 2nd: $[2,3]$
  - 3rd: $[3,4]$
  - 4th: $[4,5]$

### Step 3: Calculate the Left-Hand Riemann Sum (LHS)

For the Left-Hand Sum, use the left endpoint of each subinterval to evaluate $f(x)$:

- For $[1,2]$: $f(1) = 3(1) = 3$
- For $[2,3]$: $f(2) = 3(2) = 6$
- For $[3,4]$: $f(3) = 3(3) = 9$
- For $[4,5]$: $f(4) = 3(4) = 12$

Sum these values and multiply by the width of each subinterval ($\Delta x = 1$):

$$LHS = (3 + 6 + 9 + 12)(1) = 30$$

### Step 4: Calculate the Right-Hand Riemann Sum (RHS)

For the Right-Hand Sum, use the right endpoint of each subinterval to evaluate $f(x)$:

- For $[1,2]$: $f(2) = 3(2) = 6$
- For $[2,3]$: $f(3) = 3(3) = 9$
- For $[3,4]$: $f(4) = 3(4) = 12$
- For $[4,5]$: $f(5) = 3(5) = 15$

Sum these values and multiply by the width of each subinterval ($\Delta x = 1$):

$$RHS = (6 + 9 + 12 + 15)(1) = 42$$

### Summary

- The Left-Hand Riemann Sum for $f(x) = 3x$ over $[1,5]$ with 4 subintervals is 30.
- The Right-Hand Riemann Sum for $f(x) = 3x$ over $[1,5]$ with 4 subintervals is 42.