Evaluate the left hand and right hand sums

Mpho Ngobeni

Mpho Ngobeni

Answered question

2022-05-13

Evaluate the left hand and right hand sums for f(x)= 6-x^2 on the closed interval[-2,2] with n=4

Answer & Explanation

Nick Camelot

Nick Camelot

Skilled2023-05-13Added 164 answers

To evaluate the left-hand and right-hand sums for the function f(x) = 6 - x^2 on the closed interval [-2, 2] with n = 4, we first need to divide the interval into subintervals of equal width.
Since n = 4, we will have 4 subintervals. The width of each subinterval can be calculated by taking the difference between the endpoints and dividing by the number of subintervals. In this case, the width of each subinterval is:
Δx=ban=2(2)4=44=1
Next, we need to determine the left-hand and right-hand endpoints of each subinterval. For the left-hand sums, we choose the leftmost point of each subinterval as the height of the rectangle. For the right-hand sums, we choose the rightmost point of each subinterval.
Given that the interval is [-2, 2] and the width of each subinterval is 1, the left-hand and right-hand endpoints of the subintervals are as follows:
For the left-hand sums:
Subinterval 1: x1=2
Subinterval 2: x2=1
Subinterval 3: x3=0
Subinterval 4: x4=1
For the right-hand sums:
Subinterval 1: x1=1
Subinterval 2: x2=0
Subinterval 3: x3=1
Subinterval 4: x4=2
Now, we can calculate the left-hand sum and right-hand sum by evaluating the function at each endpoint and multiplying by the width of the subinterval.
For the left-hand sum, we have:
LH=f(x1)·Δx+f(x2)·Δx+f(x3)·Δx+f(x4)·Δx
Substituting the function f(x) = 6 - x^2 into the left-hand sum equation, we get:
LH=(6(2)2)·1+(6(1)2)·1+(602)·1+(612)·1
Simplifying the expression inside the parentheses, we have:
LH=(64)·1+(61)·1+(60)·1+(61)·1
LH=2·1+5·1+6·1+5·1
LH=2+5+6+5
LH=18
Therefore, the left-hand sum for f(x) = 6 - x^2 on the closed interval [-2, 2] with n = 4 is 18.
Now let's calculate the right-hand sum using a similar process. We have:
RH=f(x1)·Δx+f(x2)·Δx+f(x3)·Δx+f(x4)·Δx
Substituting the function f(x) = 6 - x^2 into the right-hand sum equation, we get:
RH=(6(1)2)·1+(602)·1+(612)·1+(622)·1
Simplifying the expression inside the parentheses, we have:
RH=(61)·1+(60)·1+(61)·1+(64)·1
RH=5·1+6·1+5·1+2·1
RH=5+6+5+2
RH=18
Therefore, the right-hand sum for f(x) = 6 - x^2 on the closed interval [-2, 2] with n = 4 is also 18.
In summary:
- The left-hand sum for f(x) = 6 - x^2 on the interval [-2, 2] with n = 4 is 18.
- The right-hand sum for f(x) = 6 - x^2 on the interval [-2, 2] with n = 4 is also 18.
Both the left-hand and right-hand sums provide approximations of the definite integral of the function over the given interval, using rectangles of equal width.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?