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Math  /  Calculus

QuestionGiven the function f(x)=cos⁡(πx)f(x)=\cos (\pi x), compute the right-endpoint (Riemann) sum using n=3n=3 on the interval [0,1][0,1]. R3=R_{3}= help (numbers)

Studdy Solution

STEP 1

1. The function given is f(x)=cos⁡(πx) f(x) = \cos(\pi x) .
2. We are asked to compute the right-endpoint Riemann sum with n=3 n = 3 on the interval [0,1][0, 1].
3. The interval [0,1][0, 1] is divided into n=3 n = 3 subintervals of equal width.

STEP 2

1. Determine the width of each subinterval.
2. Identify the right endpoints of each subinterval.
3. Evaluate the function at each right endpoint.
4. Compute the Riemann sum by summing the products of the function values and the subinterval width.

STEP 3

Determine the width of each subinterval:
The interval [0,1][0, 1] is divided into n=3 n = 3 subintervals. The width Δx\Delta x of each subinterval is given by:
Δx=1−03=13\Delta x = \frac{1 - 0}{3} = \frac{1}{3}

STEP 4

Identify the right endpoints of each subinterval:
The right endpoints for the subintervals are calculated as follows: - First subinterval: x1=0+Δx=13 x_1 = 0 + \Delta x = \frac{1}{3} - Second subinterval: x2=13+Δx=23 x_2 = \frac{1}{3} + \Delta x = \frac{2}{3} - Third subinterval: x3=23+Δx=1 x_3 = \frac{2}{3} + \Delta x = 1

STEP 5

Evaluate the function at each right endpoint:
f(13)=cos⁡(π⋅13)=cos⁡(π3)=12f\left(\frac{1}{3}\right) = \cos\left(\pi \cdot \frac{1}{3}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
f(23)=cos⁡(π⋅23)=cos⁡(2π3)=−12f\left(\frac{2}{3}\right) = \cos\left(\pi \cdot \frac{2}{3}\right) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}
f(1)=cos⁡(π⋅1)=cos⁡(π)=−1f(1) = \cos(\pi \cdot 1) = \cos(\pi) = -1

STEP 6

Compute the Riemann sum:
The Riemann sum R3 R_3 is given by:
R3=Δx[f(13)+f(23)+f(1)]R_3 = \Delta x \left[ f\left(\frac{1}{3}\right) + f\left(\frac{2}{3}\right) + f(1) \right]
Substitute the values:
R3=13[12+(−12)+(−1)]R_3 = \frac{1}{3} \left[ \frac{1}{2} + \left(-\frac{1}{2}\right) + (-1) \right]
Simplify the expression:
R3=13[12−12−1]R_3 = \frac{1}{3} \left[ \frac{1}{2} - \frac{1}{2} - 1 \right] R3=13×(−1)R_3 = \frac{1}{3} \times (-1) R3=−13R_3 = -\frac{1}{3}
The value of the right-endpoint Riemann sum R3 R_3 is:
−13\boxed{-\frac{1}{3}}

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