Use the Left and Right Riemann Sums with 80 rectangles to estimate the (signed) area under the curve of y=e2x−15 on the interval of [10,20].
Write your answer using the sigma notation.
To estimate the (signed) area under the curve of the function y=e^(2x)-15 on the interval [10, 20] using the Left and Right Riemann Sums with 80 rectangles, we need to break the interval into smaller subintervals and compute the areas of the rectangles formed.
First, let's determine the width of each rectangle. Since we have 80 rectangles and the interval is from 10 to 20, the width would be Δx = (20 - 10) / 80 = 0.125.
Next, we need to compute the x-values for each rectangle. For the Left Riemann Sum, we start with the left endpoint of each subinterval to determine the x-values. For the Right Riemann Sum, we start with the right endpoint. Let's denote xi as the left endpoint of each subinterval.
For the Left Riemann Sum:
xi = a + i * Δx, where i = 0, 1, 2, ..., n-1
In this case, a = 10, n = 80, and Δx = 0.125.
Now, we can compute the heights of each rectangle using the function y = e^(2x) - 15 and evaluate it at each xi point.
For the Right Riemann Sum:
xi = a + (i + 1) * Δx, where i = 0, 1, 2, ..., n-1
In this case, a = 10, n = 80, and Δx = 0.125.
Again, compute the heights of each rectangle using the function y = e^(2x) - 15 and evaluate it at each xi point.
Once we have the widths, heights, and x-values for all the rectangles, we can calculate the area of each rectangle by multiplying the width and height.
Left Riemann Sum:
The area of each rectangle is
ΔA = Δx * f(xi), where f(xi) is the evaluated height at the left endpoint xi.
The sum of all the areas would be expressed using the sigma (Σ) notation as:
Σ ΔA = Σ Δx * f(xi), where the summation is from i = 0 to n-1.
Right Riemann Sum:
The area of each rectangle is
ΔA = Δx * f(xi), where f(xi) is the evaluated height at the right endpoint xi.
The sum of all the areas would be expressed using the sigma (Σ) notation as:
Σ ΔA = Σ Δx * f(xi), where the summation is from i = 0 to n-1.
Please note that the exact calculations of the function values at each xi point and the resulting sum would require numerical methods or a computer program.