Use the Left and Right Riemann Sums with 80 rectangle to estimate the (signed) area under the curve of y=e^(3x)−5 on the interval of [10,20].
Write your answer using the sigma notation
you can confirm your work here:
http://mathworld.wolfram.com/RiemannSum.html
To estimate the (signed) area under the curve of the function y = e^(3x) - 5 on the interval [10, 20] using the Left and Right Riemann Sums, we can divide the interval into smaller subintervals and approximate the area by summing the areas of rectangles.
Step 1: Determine the width of each rectangle
To achieve a more accurate approximation, we divide the interval [10, 20] into 80 equal subintervals. The width of each rectangle can be calculated as Δx = (b - a)/n, where b is the upper limit of the interval, a is the lower limit of the interval, and n is the number of subintervals. In this case, Δx = (20 - 10)/80 = 0.125.
Step 2: Calculate the x-values of the rectangle endpoints
Since we are using the Left and Right Riemann Sums, the x-values for the rectangles in the Left Riemann Sum will be the left endpoints of each subinterval, and the x-values for the rectangles in the Right Riemann Sum will be the right endpoints of each subinterval.
For the Left Riemann Sum:
x₁ = 10
x₂ = 10 + Δx
x₃ = 10 + 2Δx
...
x₈₀ = 10 + 79Δx
For the Right Riemann Sum:
x₁ = 10 + Δx
x₂ = 10 + 2Δx
x₃ = 10 + 3Δx
...
x₈₀ = 10 + 80Δx
Step 3: Calculate the function values at each x-coordinate
Evaluate the given function y = e^(3x) - 5 at each x-coordinate obtained in Step 2. This will give us the height of each rectangle.
For the Left Riemann Sum:
f(x₁) = e^(3x₁) - 5
f(x₂) = e^(3x₂) - 5
f(x₃) = e^(3x₃) - 5
...
f(x₈₀) = e^(3x₈₀) - 5
For the Right Riemann Sum:
f(x₁) = e^(3x₁) - 5
f(x₂) = e^(3x₂) - 5
f(x₃) = e^(3x₃) - 5
...
f(x₈₀) = e^(3x₈₀) - 5
Step 4: Calculate the area for each Riemann Sum
The area of each rectangle is calculated as the width (Δx) multiplied by the height (f(x)).
For the Left Riemann Sum, the area of each rectangle is:
Δx * f(x₁) = Δx * (e^(3x₁) - 5)
Δx * f(x₂) = Δx * (e^(3x₂) - 5)
...
Δx * f(x₈₀) = Δx * (e^(3x₈₀) - 5)
For the Right Riemann Sum, the area of each rectangle is:
Δx * f(x₁) = Δx * (e^(3x₁) - 5)
Δx * f(x₂) = Δx * (e^(3x₂) - 5)
...
Δx * f(x₈₀) = Δx * (e^(3x₈₀) - 5)
Step 5: Write the answer using sigma notation
Finally, we can write the equations using sigma (Σ) notation to represent the summation of the areas of the rectangles in the Left and Right Riemann Sums.
For the Left Riemann Sum:
The (signed) area under the curve is approximately Σ[1 to 80] (Δx * (e^(3x) - 5)) where x = 10 + (i - 1)Δx
For the Right Riemann Sum:
The (signed) area under the curve is approximately Σ[1 to 80] (Δx * (e^(3x) - 5)) where x = 10 + iΔx
Please note that the actual calculation of the area requires plugging in the value of x for each rectangle and evaluating the function.