Math Problem Statement
If f(x) = 3 cos(x) 0 ≤ x ≤ 3𝜋/4 evaluate the Riemann sum with n = 6, taking the sample points to be left endpoints. (Round your answer to six decimal places.)
Solution
To evaluate the Riemann sum for the function over the interval using subintervals and left endpoints as sample points, we can follow these steps:
Step 1: Find the width of each subinterval.
The total interval is . To find the width of each subinterval, we divide the total length by :
Step 2: Identify the left endpoints of each subinterval.
The left endpoints of the subintervals will be:
Step 3: Evaluate the function at each left endpoint.
Now we evaluate at each left endpoint:
Step 4: Set up the Riemann sum.
The Riemann sum is given by the formula:
Substituting in the values:
Now, we need to compute the cosine values for the remaining points. I will do that next.
Let me calculate that quickly. math
Given values
pi = math.pi
Function f(x) = 3 * cos(x)
def f(x): return 3 * math.cos(x)
Left endpoints for x
x_values = [0, pi/8, pi/4, 3pi/8, pi/2, 5pi/8]
Calculate f(x) at each left endpoint
f_values = [f(x) for x in x_values]
Riemann sum calculation
delta_x = pi / 8 riemann_sum = delta_x * sum(f_values) riemann_sumThe Riemann sum for the function over the interval , using subintervals and left endpoints as sample points, is approximately:
This value is rounded to six decimal places.
Would you like further explanation or have any other questions?
Here are some follow-up questions that could deepen the understanding:
- How would the Riemann sum change if we used right endpoints instead of left endpoints?
- What is the difference between a left Riemann sum and a right Riemann sum?
- How does the number of subintervals () affect the accuracy of the Riemann sum?
- What is the exact value of the integral of over this interval?
- How can we approximate the area under the curve using a larger value of ?
Tip: Increasing the number of subintervals () will generally improve the accuracy of the Riemann sum, bringing it closer to the exact integral value.
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Math Problem Analysis
Mathematical Concepts
Riemann Sum
Trigonometric Functions
Cosine Function
Integral Approximation
Formulas
Riemann sum formula: S = Σ f(x_i) Δx
f(x) = 3 cos(x)
Theorems
Riemann Sum Theorem
Suitable Grade Level
Grades 11-12
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