Use the Left and Right Riemann Sums with 100 rectangle to estimate the (signed) area under the curve of y=−2x+1 on the interval [0,50]. Write your answer using the sigma notation.
To estimate the (signed) area under the curve of the function y = -2x + 1 on the interval [0, 50] using the Left and Right Riemann Sums with 100 rectangles, we need to divide the interval into equal subintervals and evaluate the function at specific points within each subinterval.
Let's start with the Left Riemann Sum. With 100 rectangles, we divide the interval [0, 50] into 100 subintervals of width Δx = (50 - 0) / 100 = 0.5.
For the Left Riemann Sum, we evaluate the function at the left endpoint of each subinterval (the function value at x = a) and multiply it by the width of the subinterval. The sum of these products gives an approximation of the area under the curve.
Using sigma notation, the Left Riemann Sum can be written as:
∑[from i = 0 to n-1] f(a + iΔx)Δx
Substituting the values, we have:
∑[from i = 0 to 99] f(0 + i0.5) * 0.5
Now, let's calculate the expression f(a + iΔx) * Δx for each i:
f(0 + 0*0.5) * 0.5 = f(0) * 0.5 = (-(0) * 2 + 1) * 0.5 = 1 * 0.5 = 0.5
f(0 + 1*0.5) * 0.5 = f(0.5) * 0.5 = (-(0.5) * 2 + 1) * 0.5 = 0.5 * 0.5 = 0.25
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f(0 + 99*0.5) * 0.5 = f(49.5) * 0.5 = (-(49.5) * 2 + 1) * 0.5 = -98 * 0.5 = -49
Now, we can add up all these values using the sigma notation:
∑[from i = 0 to 99] f(0 + i0.5) * 0.5 = 0.5 + 0.25 + ... + (-49)
To find the Right Riemann Sum, we follow the same process, but instead of evaluating the function at the left endpoint of each subinterval, we evaluate it at the right endpoint (the function value at x = a + Δx) and multiply it by the width of the subinterval.
Using sigma notation, the Right Riemann Sum can be written as:
∑[from i = 1 to n] f(a + iΔx)Δx
Substituting the values, we have:
∑[from i = 1 to 100] f(0 + i0.5) * 0.5
Now, let's calculate the expression f(a + iΔx) * Δx for each i:
f(0 + 1*0.5) * 0.5 = f(0.5) * 0.5 = (-(0.5) * 2 + 1) * 0.5 = 0.5 * 0.5 = 0.25
f(0 + 2*0.5) * 0.5 = f(1) * 0.5 = (-(1) * 2 + 1) * 0.5 = -1 * 0.5 = -0.5
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f(0 + 100*0.5) * 0.5 = f(50) * 0.5 = (-(50) * 2 + 1) * 0.5 = -99 * 0.5 = -49.5
Now, we can add up all these values using the sigma notation:
∑[from i = 1 to 100] f(0 + i0.5) * 0.5 = 0.25 + (-0.5) + ... + (-49.5)
Note that the Left Riemann Sum includes f(0), while the Right Riemann Sum includes f(a + nΔx) = f(50) in this case.
Now, you can calculate the exact numerical values for both the Left and Right Riemann Sums by evaluating these sigma notations.
To estimate the (signed) area under the curve of y = -2x + 1 on the interval [0,50] using left and right Riemann sums with 100 rectangles, we will divide the interval into 100 equal subintervals.
Let's start with the left Riemann sum.
The width of each rectangle, denoted as Δx, is calculated as:
Δx = (50 - 0) / 100 = 0.5
The left endpoints of the intervals are used to determine the height of each rectangle. We substitute the left endpoints (x-values) into the equation y = -2x + 1 to find the corresponding y-values.
The left Riemann sum can be expressed using sigma notation as:
Left Riemann Sum = Δx * Σ(-2xi + 1), for i = 0 to 99
where xi is the left endpoint of the i-th subinterval.
Now, let's calculate the right Riemann sum.
The right endpoints of the intervals are used to determine the height of each rectangle. We substitute the right endpoints (x-values) into the equation y = -2x + 1 to find the corresponding y-values.
The right Riemann sum can be expressed using sigma notation as:
Right Riemann Sum = Δx * Σ(-2xi + 1), for i = 1 to 100
where xi is the right endpoint of the i-th subinterval.
Note: Since the interval starts from 0, the left Riemann sum will have an additional term for the first subinterval, and the right Riemann sum will have an additional term for the last subinterval.
Now, using the provided interval and the width of each rectangle, we can proceed to calculate the left and right Riemann sums.