Math Problem Statement

Use this definition with right endpoints to find an expression for the area under the graph of f(x) = x^2 + √(1 + 2x) on the interval [7, 9]. Do not evaluate the limit.

Solution

To find the expression for the area under the graph of f(x)=x2+1+2xf(x) = x^2 + \sqrt{1 + 2x} on the interval [7,9][7, 9] 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 [7,9][7, 9] into nn subintervals of equal width: Δx=97n=2n.\Delta x = \frac{9 - 7}{n} = \frac{2}{n}.


Step 2: Define the right endpoints

The right endpoint of the ii-th subinterval is: xi=7+iΔx=7+i2n.x_i = 7 + i \Delta x = 7 + i \cdot \frac{2}{n}.


Step 3: Use the function value at the right endpoints

At each subinterval, evaluate f(x)f(x) at xi=7+2inx_i = 7 + \frac{2i}{n}: f(xi)=(7+2in)2+1+2(7+2in).f(x_i) = \left( 7 + \frac{2i}{n} \right)^2 + \sqrt{1 + 2\left( 7 + \frac{2i}{n} \right)}.


Step 4: Construct the Riemann sum

The Riemann sum for the area under f(x)f(x) on [7,9][7, 9] is: Riemann Sum=i=1nf(xi)Δx.\text{Riemann Sum} = \sum_{i=1}^n f(x_i) \Delta x. Substitute Δx=2n\Delta x = \frac{2}{n} and f(xi)f(x_i): Riemann Sum=i=1n[(7+2in)2+1+2(7+2in)]2n.\text{Riemann Sum} = \sum_{i=1}^n \left[ \left( 7 + \frac{2i}{n} \right)^2 + \sqrt{1 + 2\left( 7 + \frac{2i}{n} \right)} \right] \cdot \frac{2}{n}.


Step 5: Express the definite integral as a limit

The area under the graph is the limit of the Riemann sum as nn \to \infty: limni=1n[(7+2in)2+1+2(7+2in)]2n.\lim_{n \to \infty} \sum_{i=1}^n \left[ \left( 7 + \frac{2i}{n} \right)^2 + \sqrt{1 + 2\left( 7 + \frac{2i}{n} \right)} \right] \cdot \frac{2}{n}.

This is the required expression.


Would you like a detailed explanation of any step or additional examples?


Related Questions

  1. What does the Riemann sum represent geometrically?
  2. How can we calculate Δx\Delta x for any interval [a,b][a, b]?
  3. Why do we evaluate f(x)f(x) at the right endpoint and not the left?
  4. What happens to the Riemann sum as nn increases?
  5. 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