Math Problem Statement

Find the Riemann sum for f(x)=2x-1, -6<=x<=4 with five equal subintervals, taking the sample points to be right endpoints. Explain, with the aid of its graph, what the Riemann sum represents.

Solution

To calculate the Riemann sum for f(x)=2x1f(x) = 2x - 1 over the interval [6,4][-6, 4] with five equal subintervals and sample points as right endpoints, we follow these steps:


1. Determine the width of each subinterval (Δx\Delta x):

Δx=Length of intervalNumber of subintervals=4(6)5=105=2.\Delta x = \frac{\text{Length of interval}}{\text{Number of subintervals}} = \frac{4 - (-6)}{5} = \frac{10}{5} = 2.


2. Identify the right endpoints:

The subintervals are: [ [-6, -4], , [-4, -2], , [-2, 0], , [0, 2], , [2, 4]. ]

The right endpoints are: x1=4,x2=2,x3=0,x4=2,x5=4.x_1 = -4, \, x_2 = -2, \, x_3 = 0, \, x_4 = 2, \, x_5 = 4.


3. Evaluate f(x)f(x) at the right endpoints:

The function f(x)=2x1f(x) = 2x - 1 is evaluated at each right endpoint: f(4)=2(4)1=9,f(-4) = 2(-4) - 1 = -9, f(2)=2(2)1=5,f(-2) = 2(-2) - 1 = -5, f(0)=2(0)1=1,f(0) = 2(0) - 1 = -1, f(2)=2(2)1=3,f(2) = 2(2) - 1 = 3, f(4)=2(4)1=7.f(4) = 2(4) - 1 = 7.


4. Compute the Riemann sum:

The Riemann sum is: S=i=15f(xi)Δx=Δx[f(4)+f(2)+f(0)+f(2)+f(4)].S = \sum_{i=1}^{5} f(x_i) \Delta x = \Delta x \big[f(-4) + f(-2) + f(0) + f(2) + f(4)\big].

Substitute values: S=2[(9)+(5)+(1)+3+7].S = 2 \big[(-9) + (-5) + (-1) + 3 + 7\big].

Simplify: S=2×(5)=10.S = 2 \times (-5) = -10.


5. Interpretation of the Riemann sum:

The Riemann sum represents an approximation of the net area under the curve y=2x1y = 2x - 1 from x=6x = -6 to x=4x = 4, using rectangles with heights determined by the function values at the right endpoints of each subinterval. Since f(x)f(x) is linear and crosses the xx-axis, some areas are negative (where f(x)<0f(x) < 0) and others are positive (where f(x)>0f(x) > 0). The sum combines these areas, reflecting the net signed area.


Graph Illustration:

I can provide a graph showing:

  1. The line f(x)=2x1f(x) = 2x - 1 over the interval [6,4][-6, 4].
  2. The rectangles used to compute the Riemann sum.

Would you like me to generate this graph?

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Math Problem Analysis

Mathematical Concepts

Riemann Sum
Integration
Linear Functions
Area Approximation

Formulas

Riemann sum formula: S = Σ f(x_i) Δx

Theorems

Fundamental Theorem of Calculus (as related to integration approximation)

Suitable Grade Level

Grades 11-12