Math Problem Statement

Use a Riemann sum with 4 rectangles of equal width to approximate the area between y equals x cubed plus 8 and the x-axis on the interval left square bracket short dash 2 comma 1 right square bracket. Use the right-hand endpoint of each subinterval.

23.203125 space u n i t s squared

25.4953125 space u n i t s squared

16.453125 space u n i t s squared

29.42578125 space u n i t s squared

Solution

To approximate the area under the curve y=x3+8y = x^3 + 8 on the interval [2,1][-2, 1] using a Riemann sum with 4 rectangles of equal width, and using the right-hand endpoint of each subinterval, we follow these steps:

Step 1: Determine the width of each rectangle

The interval [2,1][-2, 1] has a length of: Interval Length=1(2)=3\text{Interval Length} = 1 - (-2) = 3 Since we're using 4 rectangles, the width Δx\Delta x of each rectangle is: Δx=34=0.75\Delta x = \frac{3}{4} = 0.75

Step 2: Identify the right-hand endpoints

The right-hand endpoints of the subintervals can be found by starting at the left end of the interval and adding the width of the rectangles incrementally:

  • Subinterval 1: x1=2+0.75=1.25x_1 = -2 + 0.75 = -1.25
  • Subinterval 2: x2=1.25+0.75=0.5x_2 = -1.25 + 0.75 = -0.5
  • Subinterval 3: x3=0.5+0.75=0.25x_3 = -0.5 + 0.75 = 0.25
  • Subinterval 4: x4=0.25+0.75=1x_4 = 0.25 + 0.75 = 1

Step 3: Calculate the function values at these right-hand endpoints

We need to evaluate the function y=x3+8y = x^3 + 8 at each right-hand endpoint:

  • f(1.25)=(1.25)3+8=1.953125+8=6.046875f(-1.25) = (-1.25)^3 + 8 = -1.953125 + 8 = 6.046875
  • f(0.5)=(0.5)3+8=0.125+8=7.875f(-0.5) = (-0.5)^3 + 8 = -0.125 + 8 = 7.875
  • f(0.25)=(0.25)3+8=0.015625+8=8.015625f(0.25) = (0.25)^3 + 8 = 0.015625 + 8 = 8.015625
  • f(1)=(1)3+8=1+8=9f(1) = (1)^3 + 8 = 1 + 8 = 9

Step 4: Compute the Riemann sum

Multiply each function value by the width Δx=0.75\Delta x = 0.75 and sum them up to get the Riemann sum: Riemann Sum=0.75×(6.046875+7.875+8.015625+9)\text{Riemann Sum} = 0.75 \times (6.046875 + 7.875 + 8.015625 + 9) =0.75×30.9375=23.203125 units2= 0.75 \times 30.9375 = 23.203125 \text{ units}^2

Conclusion:

The correct approximation using the Riemann sum is:

23.203125 units squared

Would you like more details on any part of the process or have any other questions?

Here are 5 related questions you might find helpful:

  1. How do you compute a left-hand Riemann sum?
  2. What is the difference between a Riemann sum and an integral?
  3. How does increasing the number of rectangles in a Riemann sum affect the accuracy?
  4. Can you explain the trapezoidal rule as an alternative to Riemann sums?
  5. How do you set up a Riemann sum for a different function and interval?

Tip: When using Riemann sums, increasing the number of rectangles or using more sophisticated methods like the trapezoidal rule can lead to more accurate approximations of the area under a curve.

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Math Problem Analysis

Mathematical Concepts

Calculus
Riemann Sum
Area Approximation

Formulas

Riemann Sum formula

Theorems

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Suitable Grade Level

Advanced High School