Consider the given function.
f(x) = 3 sin(6x) text(, ) 0<=x<=(3pi)/2
(a) Find the Riemann sum with six terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.)
R6 =
(b) Repeat part (a) with midpoints as sample points. (Give your answer correct to six decimal places.)
M6 =
Divide your integration domain by 6
delta x = [(3 pi/2) - 0] / 6
delta x = pi/4
Label your 7 X values
Xo = 0 , X1 = pi/4, X2= pi/2, X3 = 3 pi/4, X4 = pi, X5 = 5 pi/4, X6 = 3 pi/2
calculate f(x) at X1 through X6
(You do not need f(Xo) because you are taking right side of rectangles not left side)
Add those 6 values of f(x) and multiply by delta x
the end
( I told you how to do midpoints in the other problem )
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To find the Riemann sum with right endpoints, we need to divide the interval [0, (3π)/2] into six subintervals of equal width. Then, we evaluate the function at the right endpoint of each subinterval and multiply it by the width of the subinterval.
(a) Riemann sum with right endpoints:
Step 1: Find the width of each subinterval.
The total width of the interval [0, (3π)/2] is (3π)/2 - 0 = (3π)/2. Since we want to divide it into six subintervals, the width of each subinterval is ((3π)/2)/6 = π/4.
Step 2: Calculate the right endpoints.
Starting from 0, the right endpoints of the subintervals are π/4, π/2, 3π/4, π, 5π/4, and 3π/2.
Step 3: Evaluate the function at the right endpoints.
Evaluate f(x) = 3sin(6x) at each right endpoint:
f(π/4) = 3sin(6(π/4)) = 3sin(3π/2) = 3(-1) = -3
f(π/2) = 3sin(6(π/2)) = 3sin(3π) = 3sin(π) = 3(0) = 0
f(3π/4) = 3sin(6(3π/4)) = 3sin(9π/2) = 3(-1) = -3
f(π) = 3sin(6(π)) = 3sin(6π) = 3sin(0) = 3(0) = 0
f(5π/4) = 3sin(6(5π/4)) = 3sin(15π/4) = 3(1) = 3
f(3π/2) = 3sin(6(3π/2)) = 3sin(9π) = 3sin(π) = 3(0) = 0
Step 4: Calculate the Riemann sum.
Multiply each function value by the width of the subinterval and add them up.
R6 = (π/4)(-3) + (π/4)(0) + (π/4)(-3) + (π/4)(0) + (π/4)(3) + (π/4)(0)
= -3π/4 + 0 + -3π/4 + 0 + 3π/4 + 0
= -6π/4 + 3π/4
= -3π/4
≈ -2.356194
Therefore, the Riemann sum with six terms, taking the sample points to be right endpoints, is approximately -2.356194.
(b) Riemann sum with midpoints:
To find the Riemann sum with midpoints, we need to divide the interval [0, (3π)/2] into six subintervals of equal width. Then, we evaluate the function at the midpoint of each subinterval and multiply it by the width of the subinterval.
Step 1: Find the width of each subinterval.
The width of each subinterval is still π/4, as we divided the interval [0, (3π)/2] into six equal parts.
Step 2: Calculate the midpoints.
Starting from 0, the midpoints of the subintervals are π/8, 3π/8, 5π/8, 7π/8, 9π/8, and 11π/8.
Step 3: Evaluate the function at the midpoints.
Evaluate f(x) = 3sin(6x) at each midpoint:
f(π/8) = 3sin(6(π/8)) = 3sin(3π/4) = 3(1/√2) = 3/√2
f(3π/8) = 3sin(6(3π/8)) = 3sin(9π/4) = 3(-1/√2) = -3/√2
f(5π/8) = 3sin(6(5π/8)) = 3sin(15π/8) = 3(1/√2) = 3/√2
f(7π/8) = 3sin(6(7π/8)) = 3sin(21π/8) = 3(-1/√2) = -3/√2
f(9π/8) = 3sin(6(9π/8)) = 3sin(27π/8) = 3(1/√2) = 3/√2
f(11π/8) = 3sin(6(11π/8)) = 3sin(33π/8) = 3(-1/√2) = -3/√2
Step 4: Calculate the Riemann sum.
Multiply each function value by the width of the subinterval and add them up.
M6 = (π/4)((3/√2) - (3/√2) + (3/√2) - (3/√2) + (3/√2) - (3/√2))
= (π/4)(0)
= 0
Therefore, the Riemann sum with six terms, taking the sample points to be midpoints, is 0.