Use the Left and Right Riemann Sums with 100 rectangles to estimate the (signed) area under the curve of y=−6x+8 on the interval [0,50]. Write your answer using the sigma notation.
∆x = 50/100 = 1/2
so the Left Sum area is the sum of a lot of rectangles of height f(x) and width ∆x
99
∑ f(k*∆x)*∆x = ∑(-6k∆x+8)*∆x = ∑(-6k/2+8)(1/2) = ∑(-3k/2 + 8)
0
For the Right sum, sum over k=1..100 instead
To estimate the (signed) area under the curve of y = -6x + 8 on the interval [0,50] using the Left and Right Riemann Sums with 100 rectangles, we need to divide the interval [0,50] into 100 subintervals of equal width.
The width of each rectangle (Δx) is equal to the interval width divided by the number of rectangles:
Δx = (50 - 0) / 100 = 0.5
Let's start with the Left Riemann Sum. For each rectangle, we will use the left endpoint to calculate the height of the rectangle.
The left endpoint of the k-th rectangle is given by:
x_k = (k - 1) * Δx
where k represents the rectangle number.
The height of the rectangle is given by evaluating the function at x_k:
y_k = -6 * x_k + 8
= -6 * ((k - 1) * Δx) + 8
= -6 * (k - 1) * 0.5 + 8
The area of the k-th rectangle is the product of the height and the width:
A_k = y_k * Δx
To estimate the (signed) area using the Left Riemann Sum, we sum up the areas of all the rectangles:
A_Left = Σ(A_k)
= Σ((-6 * (k - 1) * 0.5 + 8) * 0.5)
where the summation is taken over k from 1 to 100.
Similarly, for the Right Riemann Sum, the right endpoint of the k-th rectangle is given by:
x_k = k * Δx
The height of the rectangle is given by evaluating the function at x_k:
y_k = -6 * x_k + 8
= -6 * (k * Δx) + 8
= -6 * k * 0.5 + 8
The area of the k-th rectangle is the product of the height and the width:
A_k = y_k * Δx
To estimate the (signed) area using the Right Riemann Sum, we sum up the areas of all the rectangles:
A_Right = Σ(A_k)
= Σ((-6 * k * 0.5 + 8) * 0.5)
where the summation is taken over k from 1 to 100.
You can simplify the sigma notation and calculate the values of A_Left and A_Right using the appropriate formulas.