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 on the interval 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 has a length of: Since we're using 4 rectangles, the width of each rectangle is:
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:
- Subinterval 2:
- Subinterval 3:
- Subinterval 4:
Step 3: Calculate the function values at these right-hand endpoints
We need to evaluate the function at each right-hand endpoint:
Step 4: Compute the Riemann sum
Multiply each function value by the width and sum them up to get the Riemann sum:
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:
- How do you compute a left-hand Riemann sum?
- What is the difference between a Riemann sum and an integral?
- How does increasing the number of rectangles in a Riemann sum affect the accuracy?
- Can you explain the trapezoidal rule as an alternative to Riemann sums?
- 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
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