Math Problem Statement
Find the value
V
of the Riemann sum
V
n ∑ k
1
f ( c k ) Δ x k
for the function
f
(
x
)
x
2
−
2
using the partition
P
{
1
,
2
,
3
,
5
}
, where
c
k
is the left endpoints of the partition.
Solution
To find the value of the Riemann sum for the function using the partition with left endpoints, we need to compute the sum:
where represents the left endpoints of the subintervals, and represents the width of each subinterval.
Step-by-Step Solution
-
Identify the subintervals and their widths :
The partition creates the following subintervals:
Therefore, the widths of the subintervals are:
-
Determine the left endpoints for each subinterval:
Since we are using left endpoints:
- For the interval ,
- For the interval ,
- For the interval ,
-
Evaluate for each left endpoint:
The function is . Let's compute :
-
Compute the Riemann sum :
Now, we calculate the Riemann sum using the formula:
Substitute the values we found:
Simplify the expression:
Conclusion
The value of the Riemann sum for the function using the partition with left endpoints is 15.
Would you like more details on any of these steps, or do you have any questions?
Here are 5 related questions to consider:
- What would be the value of the Riemann sum if we used right endpoints instead of left endpoints?
- How would the Riemann sum change if the function was ?
- Can you compute the Riemann sum for a uniform partition with on the interval ?
- How does the choice of partition affect the accuracy of the Riemann sum approximation for the integral of a function?
- What is the difference between a left Riemann sum, a right Riemann sum, and a midpoint Riemann sum?
Tip: When working with Riemann sums, always carefully consider the choice of endpoints and the width of each subinterval, as they directly impact the accuracy of the approximation for the integral.
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Math Problem Analysis
Mathematical Concepts
Riemann Sum
Partitioning Intervals
Functions and Limits
Formulas
Riemann Sum Formula: V = ∑ f(ck)Δxk
Function: f(x) = x^2 - 2
Theorems
Riemann Sum Approximation for Definite Integrals
Suitable Grade Level
Grades 11-12 or early college
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