1. What is the Right Riemann Sum for f(x)=x−3 over the interval [8,11], with n=3 rectangles of equal width?
2. What is the right Riemann sum of f(x)=x+3 when using 3 rectangles in the range [−3,−1]?
3. What is the left Riemann sum of f(x)=−2x+1 when using 2 rectangles in the range [−2,0]?
4. What is the left Riemann sum of f(x)=−3x+2 when using 4 rectangles of equal width in the interval [2,4]?
since the right side of the intervals are at 9,10,11 just add up (1)* f(9)+f(10)+f(11)
Again use the right sides, of width 2/3 each, to get (2/3)*(f(-7/3)+f(-5/3)+f(-1))
Do the left sums similarly, using the appropriate widths.
To find the Right Riemann Sum for a given function over a specific interval with a given number of rectangles, you need to follow a few steps.
1. Determine the width of each rectangle:
- The width of each rectangle is calculated by dividing the length of the interval by the number of rectangles. In this case, the interval is given as [a, b], and the number of rectangles is n. So, the width of each rectangle is (b - a) / n.
2. Identify the x-values of the right endpoints of each rectangle:
- For the Right Riemann Sum, you use the right endpoints of each rectangle. The x-value of the right endpoint of the first rectangle is the lower limit of the interval plus one rectangle width. The x-value of the right endpoint of the second rectangle is the lower limit of the interval plus two rectangle widths, and so on.
3. Evaluate the function at each right endpoint:
- Once you have the x-values of the right endpoints, you substitute these values into the given function to find the corresponding y-values.
4. Calculate the area of each rectangle:
- The area of each rectangle is determined by multiplying the width of the rectangle by the corresponding y-value from the function evaluation.
5. Sum up the areas of all rectangles:
- Finally, you add up the areas of all the rectangles to find the Riemann Sum.
Let's apply these steps to each of the given questions:
1. The width of each rectangle is (11 - 8) / 3 = 1.
- The x-values of the right endpoints are 9, 10, and 11.
- Evaluating the function at these points:
- f(9) = 9 - 3 = 6
- f(10) = 10 - 3 = 7
- f(11) = 11 - 3 = 8
- The areas of the rectangles are 1 * 6, 1 * 7, and 1 * 8.
- The Right Riemann Sum is 6 + 7 + 8 = 21.
2. The width of each rectangle is (-1 - (-3)) / 3 = 2 / 3.
- The x-values of the right endpoints are -2 + (2/3), -2 + (4/3), and -2 + 2.
- Evaluating the function at these points:
- f(-2 + (2/3)) = -2 + (2/3) + 3 = -2 + 2 + 3 = 3.33
- f(-2 + (4/3)) = -2 + (4/3) + 3 = -2 + 4/3 + 3 = 4
- f(-2 + 2) = -2 + 2 + 3 = 3
- The areas of the rectangles are (2/3) * 3.33, (2/3) * 4, and (2/3) * 3.
- The Right Riemann Sum is 2.22 + 2.67 + 2 = 7.89.
3. The width of each rectangle is (0 - (-2)) / 2 = 1.
- The x-values of the right endpoints are -2 + 1 and -2 + 2.
- Evaluating the function at these points:
- f(-2 + 1) = -2(1) + 1 = -1
- f(-2 + 2) = -2(2) + 1 = -3
- The areas of the rectangles are 1 * (-1) and 1 * (-3).
- The Left Riemann Sum is (-1) + (-3) = -4.
4. The width of each rectangle is (4 - 2) / 4 = 0.5.
- The x-values of the right endpoints are 2.5, 3, 3.5, and 4.
- Evaluating the function at these points:
- f(2.5) = -3(2.5) + 2 = -7.5 + 2 = -5.5
- f(3) = -3(3) + 2 = -9 + 2 = -7
- f(3.5) = -3(3.5) + 2 = -10.5 + 2 = -8.5
- f(4) = -3(4) + 2 = -12 + 2 = -10
- The areas of the rectangles are 0.5 * (-5.5), 0.5 * (-7), 0.5 * (-8.5), and 0.5 * (-10).
- The Left Riemann Sum is (-2.75) + (-3.5) + (-4.25) + (-5) = -15.5.
I hope these explanations help you understand how to find the Riemann Sums for different functions and intervals using various numbers of rectangles.