← Back to learning objective

Right Riemann sum with n = 5

Consider the function f(x)=xsin⁡(x)f(x) = x\sin(x) on the interval [0,π][0, \pi].

Partition [0,π][0, \pi] into n=5n = 5 equal subintervals and use the right Riemann sum to approximate ∫0πxsin⁡(x) dx\displaystyle\int_0^{\pi} x\sin(x)\, dx. Compute the value of this right Riemann sum.

Right Riemann sum with n = 5 graph
Hint

Compute Δx=π/5\Delta x = \pi/5, then evaluate ff at the five right endpoints x1,x2,…,x5=π/5,2π/5,…,πx_1, x_2, \dots, x_5 = \pi/5, 2\pi/5, \dots, \pi. Sum Δx⋅f(xi)\Delta x \cdot f(x_i) over these values, being careful with radians when evaluating sine.

Step-by-step solution

Since [0,π][0,\pi] is divided into n=5n=5 equal subintervals, the width is Δx=π/5≈0.6283\Delta x = \pi/5 \approx 0.6283. The right endpoints are xi=iπ/5x_i = i\pi/5 for i=1,2,3,4,5i = 1, 2, 3, 4, 5, giving x1=π/5,x2=2π/5,x3=3π/5,x4=4π/5,x5=πx_1 = \pi/5, x_2 = 2\pi/5, x_3 = 3\pi/5, x_4 = 4\pi/5, x_5 = \pi.

Evaluating f(x)=xsin⁡(x)f(x) = x\sin(x) at each right endpoint: f(π/5)≈0.6283(0.5878)≈0.3693f(\pi/5) \approx 0.6283(0.5878) \approx 0.3693, f(2π/5)≈1.2566(0.9511)≈1.1955f(2\pi/5) \approx 1.2566(0.9511) \approx 1.1955, f(3π/5)≈1.8850(0.9511)≈1.7924f(3\pi/5) \approx 1.8850(0.9511) \approx 1.7924, f(4π/5)≈2.5133(0.5878)≈1.4775f(4\pi/5) \approx 2.5133(0.5878) \approx 1.4775, f(π)≈3.1416(0)=0f(\pi) \approx 3.1416(0) = 0. Summing these and multiplying by Δx≈0.6283\Delta x \approx 0.6283 gives the right Riemann sum R5≈0.6283(0.3693+1.1955+1.7924+1.4775+0)≈0.6283(4.8347)≈3.037R_5 \approx 0.6283(0.3693 + 1.1955 + 1.7924 + 1.4775 + 0) \approx 0.6283(4.8347) \approx 3.037.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap