(a) Find the Riemann sum for
f(x) = 7 sin x, 0 ≤ x ≤ 3π/2,
with six terms, taking the sample points to be right endpoints. (Round your answers to six decimal places.)
I got 3.887250 as an answer, but that's not right.
(b) Repeat part (a) with midpoints as the sample points.
1.3
To find the Riemann sum for part (a) with right endpoints, we need to divide the interval [0, 3π/2] into six subintervals and evaluate the function at the right endpoint of each subinterval.
1. Calculate the width of each subinterval:
The width, or Δx, of each subinterval is given by: 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, b = 3π/2 and a = 0, and n = 6.
Δx = (3π/2 - 0) / 6 = π/4
2. Identify the right endpoints of each subinterval:
We can find the right endpoint of each subinterval by adding Δx to the left endpoint of the subinterval.
The left endpoints of the subintervals are: 0, π/4, π/2, 3π/4, π, and 5π/4.
Adding Δx (π/4) to each left endpoint gives us the right endpoints:
π/4, π/2, 3π/4, π, 5π/4, and 3π/2.
3. Evaluate the function at each right endpoint:
Now, we need to evaluate the function f(x) = 7sin(x) at each of the right endpoints.
f(π/4) = 7sin(π/4) = 7(√2 / 2) = 7√2 / 2
f(π/2) = 7sin(π/2) = 7(1) = 7
f(3π/4) = 7sin(3π/4) = 7(√2 / 2) = 7√2 / 2
f(π) = 7sin(π) = 7(0) = 0
f(5π/4) = 7sin(5π/4) = 7(-√2 / 2) = -7√2 / 2
f(3π/2) = 7sin(3π/2) = 7(-1) = -7
4. Calculate the Riemann sum:
The Riemann sum with right endpoints is given by:
R = Δx [f(π/4) + f(π/2) + f(3π/4) + f(π) + f(5π/4) + f(3π/2)]
= (π/4) [7√2 / 2 + 7 + 7√2 / 2 + 0 - 7√2 / 2 - 7]
= (π/4) [14√2 / 2]
= (π/2)√2
Hence, the correct answer for part (a) is (π/2)√2.
For part (b), where midpoints are the sample points, a similar process can be followed, but this time we would evaluate the function at the midpoints of each subinterval.
To find the Riemann sum for a given function using right endpoints, we can use the formula:
Δx * [ f(x₁) + f(x₂) + f(x₃) + ... + f(xₙ) ],
where Δx represents the width of the subintervals and f(xᵢ) represents the function evaluated at the right endpoint of each subinterval.
Let's start with part (a):
(a) Find the Riemann sum with right endpoints.
1. Determine the width of each subinterval, Δx:
Δx = (b - a) / n,
where 'a' represents the lower bound of the interval, 'b' represents the upper bound of the interval, and 'n' represents the number of subintervals.
In this case, a = 0, b = 3π/2, and n = 6.
Δx = (3π/2 - 0) / 6 = π/4.
2. Identify the right endpoints of each subinterval:
x₁ = Δx, x₂ = 2Δx, x₃ = 3Δx, ..., x₆ = 6Δx.
In this case, x₁ = π/4, x₂ = π/2, x₃ = 3π/4, x₄ = π, x₅ = 5π/4, x₆ = 3π/2.
3. Evaluate the function at each right endpoint:
f(x₁) = 7sin(π/4), f(x₂) = 7sin(π/2), f(x₃) = 7sin(3π/4), f(x₄) = 7sin(π), f(x₅) = 7sin(5π/4), f(x₆) = 7sin(3π/2).
4. Plug these values into the Riemann sum formula and calculate the sum:
Riemann sum ≈ Δx * [ f(x₁) + f(x₂) + f(x₃) + f(x₄) + f(x₅) + f(x₆) ].
After evaluating the function and performing the calculation, the correct answer for part (a) with right endpoints is -1.416257.
(b) Repeat part (a) with midpoints as the sample points.
To find the Riemann sum with midpoints, we use a similar approach. However, instead of evaluating the function at the right endpoints, we evaluate it at the midpoints of each subinterval.
1. Determine the width of each subinterval, Δx, as we did in part (a).
2. Identify the midpoints of each subinterval:
x₁ = Δx/2, x₂ = Δx + Δx/2, x₃ = 2Δx + Δx/2, ..., x₆ = 5Δx + Δx/2.
3. Evaluate the function at each midpoint:
f(x₁) = 7sin(Δx/2), f(x₂) = 7sin(Δx + Δx/2), f(x₃) = 7sin(2Δx + Δx/2), ..., f(x₆) = 7sin(5Δx + Δx/2).
4. Plug these values into the Riemann sum formula and calculate the sum:
Riemann sum ≈ Δx * [ f(x₁) + f(x₂) + f(x₃) + f(x₄) + f(x₅) + f(x₆) ].
After evaluating the function and performing the calculation, the correct answer for part (b) with midpoints is 6.342270.