← Back to learning objective

Left Riemann sum with n = 4

Consider the function f(x)=x2+1f(x) = x^2 + 1 on the interval [0,4][0, 4].

Partition [0,4][0, 4] into n=4n = 4 equal subintervals and use the left endpoint of each subinterval to approximate the area under f(x)=x2+1f(x) = x^2 + 1 on [0,4][0, 4]. Compute the left Riemann sum.

Hint

First find Δx=4−04\Delta x = \frac{4-0}{4}, then list the left endpoints x0,x1,x2,x3x_0, x_1, x_2, x_3 of the four subintervals. Compute ff at each left endpoint, sum the results, and multiply the total by Δx\Delta x.

Step-by-step solution

With n=4n = 4 subintervals on [0,4][0,4], the width is Δx=4−04=1\Delta x = \frac{4-0}{4} = 1. This gives subintervals [0,1],[1,2],[2,3],[3,4][0,1], [1,2], [2,3], [3,4] with left endpoints x0=0,x1=1,x2=2,x3=3x_0=0, x_1=1, x_2=2, x_3=3.

Evaluating ff at each left endpoint: f(0)=1f(0) = 1, f(1)=2f(1) = 2, f(2)=5f(2) = 5, f(3)=10f(3) = 10. The left Riemann sum is L4=Δx[f(0)+f(1)+f(2)+f(3)]=1⋅(1+2+5+10)=18.L_4 = \Delta x \left[f(0)+f(1)+f(2)+f(3)\right] = 1\cdot(1+2+5+10) = 18.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap