Math Problem Statement
Solution
To find the expression for the area under the graph of on the interval using the definition of the definite integral with right endpoints, follow these steps:
Step 1: Partition the interval and determine the width of each subinterval
Divide the interval into subintervals of equal width:
Step 2: Define the right endpoints
The right endpoint of the -th subinterval is:
Step 3: Use the function value at the right endpoints
At each subinterval, evaluate at :
Step 4: Construct the Riemann sum
The Riemann sum for the area under on is: Substitute and :
Step 5: Express the definite integral as a limit
The area under the graph is the limit of the Riemann sum as :
This is the required expression.
Would you like a detailed explanation of any step or additional examples?
Related Questions
- What does the Riemann sum represent geometrically?
- How can we calculate for any interval ?
- Why do we evaluate at the right endpoint and not the left?
- What happens to the Riemann sum as increases?
- Can we rewrite this sum for midpoint Riemann sums instead?
Tip
Always carefully define the endpoints and interval partition to avoid errors when setting up a Riemann sum.
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Math Problem Analysis
Mathematical Concepts
Definite Integration
Riemann Sum
Limits
Functions
Formulas
Δx = (b - a) / n
x_i = a + iΔx
Riemann Sum = Σ f(x_i)Δx
Definite Integral = lim(n → ∞) Σ f(x_i)Δx
Theorems
Definition of Definite Integral
Limit of Riemann Sums
Suitable Grade Level
Grades 11-12 or College-level Calculus
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