Math Problem Statement
Solve question 22
Solution
Let’s solve question 22 step by step.
Problem Restatement:
The function is defined on the interval . We are tasked with:
- Showing that the left sum for the partition of the interval is given by:
- Using the summation formula: to simplify to the form:
- Computing the limit of as .
Step 1: Left Sum Formula
The interval is divided into subintervals of equal width:
The left-hand sum is calculated as: where represents the left endpoint of the -th subinterval. For this problem:
Substituting into the formula:
Simplify the expression:
Thus, we have shown:
Step 2: Simplifying Using the Summation Formula
Using the given summation formula: we substitute this into :
Simplify:
Thus, the simplified expression for is:
Step 3: Compute the Limit of as
To find: use the expression:
Expand :
Substitute back:
Split into terms:
Simplify:
Take the limit as :
Final Answer:
The limit of as is:
Would you like a detailed explanation of any specific step, or do you have further questions? Here are related questions for practice:
- How is the left-hand sum formula derived for general functions?
- How can you compute the right-hand sum for the same function ?
- How does the summation formula for relate to the derivation of ?
- What is the integral of over , and how does it compare to ?
- How do Riemann sums converge to the definite integral?
Tip: When simplifying expressions with limits, always consider factoring terms to isolate dominant terms for large .
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Riemann Sums
Summation
Limits
Definite Integrals
Formulas
L_n = (1/n^3) * Σ(k=0 to n-1) k^2
Summation formula: Σ(k=0 to n-1) k^2 = (n-1)n(2n-1)/6
Expansion: (n-1)(2n-1) = 2n^2 - 3n + 1
Limit as n -> ∞: lim(L_n) = 1/3
Theorems
Summation Formula for k^2
Riemann Sum Definition
Suitable Grade Level
Grades 11-12 (Advanced Calculus/Pre-University)