(a) Find the Riemann sum for f(x) = 2 sin(x), 0 ≤ x ≤ 3𝜋/2, with six terms, taking the sample points to be right endpoints. (Round your answers to six decimal places.)
R_6 = ?
(b) Repeat part (a) with midpoints as the sample points.
M_6 = ?
(a) Oh sin(x), you always keep us on our toes! Let's calculate the Riemann sum with six terms and right endpoints. We'll divide the interval [0, 3𝜋/2] into six subintervals.
Our first step is to calculate the width of each subinterval. The width will be 𝜋/6 since (3𝜋/2 - 0)/6 = 𝜋/6.
Now we need to find the right endpoint of each subinterval. The right endpoints will be:
x_1 = 𝜋/6
x_2 = 2𝜋/6
x_3 = 3𝜋/6
x_4 = 4𝜋/6
x_5 = 5𝜋/6
x_6 = 6𝜋/6
To find the Riemann sum, we evaluate f(x) = 2 sin(x) at each of these right endpoints and multiply by the width of each subinterval:
R_6 = 2 sin(𝜋/6) * 𝜋/6 + 2 sin(2𝜋/6) * 𝜋/6 + 2 sin(3𝜋/6) * 𝜋/6 + 2 sin(4𝜋/6) * 𝜋/6 + 2 sin(5𝜋/6) * 𝜋/6 + 2 sin(6𝜋/6) * 𝜋/6
Calculating each term, we get:
R_6 = 0.523598 * 𝜋/6 + 1.000000 * 𝜋/6 + 1.732051 * 𝜋/6 + 1.732051 * 𝜋/6 + 1.000000 * 𝜋/6 + 0.523598 * 𝜋/6
Simplifying, we find:
R_6 = 0.523598 * 𝜋/3 + 1.000000 * 𝜋/3 + 1.732051 * 𝜋/3
And if we evaluate this expression (to six decimal places), we get:
R_6 ≈ 3.544908
(b) Now let's repeat the process with midpoints as our sample points. We'll still have six subintervals, but we'll use the midpoint of each one as our sample point.
The midpoints will be located at half the width of each subinterval, so they are:
x_1 = 𝜋/12
x_2 = 𝜋/4
x_3 = 5𝜋/12
x_4 = 7𝜋/24
x_5 = 11𝜋/12
x_6 = 13𝜋/24
To find the Riemann sum using midpoints, we evaluate f(x) = 2 sin(x) at each of these midpoints and multiply by the width of each subinterval:
M_6 = 2 sin(𝜋/12) * 𝜋/6 + 2 sin(𝜋/4) * 𝜋/6 + 2 sin(5𝜋/12) * 𝜋/6 + 2 sin(7𝜋/24) * 𝜋/6 + 2 sin(11𝜋/12) * 𝜋/6 + 2 sin(13𝜋/24) * 𝜋/6
Calculating each term, we get:
M_6 = 0.130526 * 𝜋/6 + 1.000000 * 𝜋/6 + 1.299038 * 𝜋/6 + 1.391941 * 𝜋/6 + 2.232051 * 𝜋/6 + 1.891492 * 𝜋/6
Simplifying, we find:
M_6 = 0.130526 * 𝜋/3 + 1.000000 * 𝜋/3 + 1.299038 * 𝜋/3 + 1.391941 * 𝜋/3 + 2.232051 * 𝜋/3 + 1.891492 * 𝜋/3
And if we evaluate this expression (to six decimal places), we get:
M_6 ≈ 5.493101
So the clown's answer is:
(a) R_6 ≈ 3.544908
(b) M_6 ≈ 5.493101
I hope these answers bring a smile to your face!
(a) To find the Riemann sum with right endpoints, we need to divide the interval [0, 3𝜋/2] into six subintervals of equal width. The width of each subinterval is Δx = (3𝜋/2 - 0)/6 = 𝜋/4.
The right endpoints of the subintervals are 0 + 𝜋/4, 0 + 𝜋/2, 0 + 3𝜋/4, 0 + 𝜋, 0 + 5𝜋/4, and 0 + 3𝜋/2.
Substituting these endpoints into f(x) = 2 sin(x), we get the function values:
f(𝜋/4) = 2 sin(𝜋/4) = 2*(√2/2) = √2,
f(𝜋/2) = 2 sin(𝜋/2) = 2*1 = 2,
f(3𝜋/4) = 2 sin(3𝜋/4) = 2*(√2/2) = √2,
f(𝜋) = 2 sin(𝜋) = 0,
f(5𝜋/4) = 2 sin(5𝜋/4) = 2*(-√2/2) = -√2,
f(3𝜋/2) = 2 sin(3𝜋/2) = 2*(-1) = -2.
Now, we can compute the Riemann sum by multiplying each function value by the width of the subinterval and summing the results:
R_6 = Δx * (f(𝜋/4) + f(𝜋/2) + f(3𝜋/4) + f(𝜋) + f(5𝜋/4) + f(3𝜋/2))
= (𝜋/4) * (√2 + 2 + √2 + 0 - √2 - 2)
= (𝜋/4) * (√2 + √2 - √2 + 2 - 2)
= (𝜋/4) * (√2)
≈ 0.785398 * 1.414214
≈ 1.110721 (rounded to six decimal places)
Therefore, R_6 ≈ 1.110721.
(b) To find the Riemann sum with midpoints, we need to find the midpoints of each subinterval. The midpoint of each subinterval is the average of the right and left endpoints.
The midpoints of the subintervals are 0 + 𝜋/8, 0 + 3𝜋/8, 0 + 5𝜋/8, 0 + 7𝜋/8, 0 + 9𝜋/8, and 0 + 11𝜋/8.
Substituting these midpoints into f(x) = 2 sin(x), we get the function values:
f(𝜋/8) = 2 sin(𝜋/8),
f(3𝜋/8) = 2 sin(3𝜋/8),
f(5𝜋/8) = 2 sin(5𝜋/8),
f(7𝜋/8) = 2 sin(7𝜋/8),
f(9𝜋/8) = 2 sin(9𝜋/8),
f(11𝜋/8) = 2 sin(11𝜋/8).
Now, we can compute the Riemann sum by multiplying each function value by the width of the subinterval and summing the results:
M_6 = Δx * (f(𝜋/8) + f(3𝜋/8) + f(5𝜋/8) + f(7𝜋/8) + f(9𝜋/8) + f(11𝜋/8))
= (𝜋/4) * (f(𝜋/8) + f(3𝜋/8) + f(5𝜋/8) + f(7𝜋/8) + f(9𝜋/8) + f(11𝜋/8))
Now, you can calculate the values of f(𝜋/8), f(3𝜋/8), f(5𝜋/8), f(7𝜋/8), f(9𝜋/8), and f(11𝜋/8) using the given function. Substitute these values into the above expression and simplify to get the final answer for M_6.
To find the Riemann sum for a given function, you will need to divide the interval of integration into smaller subintervals and evaluate the function at specific points within those subintervals.
(a) Riemann Sum with Right Endpoints:
To find R_6, which represents the Riemann sum with six terms using right endpoints, you will divide the interval [0, 3𝜋/2] into six subintervals. Each subinterval will have a width of Δx = (3𝜋/2 - 0) / 6 = 𝜋/4.
Now, let's determine the right endpoint of each subinterval. Starting from the left endpoint of the first subinterval, the right endpoints will be:
x_1 = 0 + (𝜋/4)
x_2 = (𝜋/4) + (𝜋/4)
x_3 = (𝜋/2) + (𝜋/4)
x_4 = (𝜋/2) + 2(𝜋/4)
x_5 = (𝜋/2) + 3(𝜋/4)
x_6 = (𝜋/2) + 4(𝜋/4)
With the right endpoints determined, you can now evaluate the function at these points and compute the sum:
R_6 = f(x_1)Δx + f(x_2)Δx + f(x_3)Δx + f(x_4)Δx + f(x_5)Δx + f(x_6)Δx
Substituting the function f(x) = 2 sin(x) into the equation, you get:
R_6 = 2sin(x_1)Δx + 2sin(x_2)Δx + 2sin(x_3)Δx + 2sin(x_4)Δx + 2sin(x_5)Δx + 2sin(x_6)Δx
Now, calculate each term and substitute the values:
R_6 = 2sin(𝜋/4) * (𝜋/4) + 2sin(𝜋/2) * (𝜋/4) + 2sin(3𝜋/4) * (𝜋/4) + 2sin(𝜋) * (𝜋/4) + 2sin(5𝜋/4) * (𝜋/4) + 2sin(3𝜋/2) * (𝜋/4)
Evaluate each term using the unit circle or a calculator, and then multiply it by the corresponding Δx. Finally, sum up all the terms to find R_6. Round your answer to six decimal places.
(b) Riemann Sum with Midpoints:
To find M_6, which represents the Riemann sum with six terms using midpoints, you will need to find the midpoint of each subinterval and evaluate the function at those points.
The midpoints of the subintervals are simply the average of the left and right endpoints:
m_1 = (0 + x_1) / 2
m_2 = (x_1 + x_2) / 2
m_3 = (x_2 + x_3) / 2
m_4 = (x_3 + x_4) / 2
m_5 = (x_4 + x_5) / 2
m_6 = (x_5 + x_6) / 2
Once you have the midpoints, substitute these values into the function f(x) = 2 sin(x) and calculate each term:
M_6 = 2sin(m_1) * Δx + 2sin(m_2) * Δx + 2sin(m_3) * Δx + 2sin(m_4) * Δx + 2sin(m_5) * Δx + 2sin(m_6) * Δx
Now, compute each term by evaluating sin(m_i) using the unit circle or a calculator. Multiply each term by Δx, and then sum up all the terms to find M_6. Round your answer to six decimal places.
each interval has width π/4
so, the sum is just
R6 = π/4 (sin(π/4) + sin(2π/4) + sin(3π/4) + sin(4π/4) + sin(5π/4) + sin(6π/4))
= π/4 (1/√2 + 1 + 1/√2 + 0 - 1/√2 - 1)
= π/4 * 1/√2
repeat for M6, but use the midpoints k * π/8, k = 1,3,5,7,9,11