the graph of a function f is given.\nestimate $int_{0}^{10}f(x)dx$ using five subintervals with the…

Explanation:

Step1: Determine the width of sub - intervals

The interval is ([0,10]) and (n = 5). The width (\Delta x=\frac{b - a}{n}=\frac{10-0}{5}=2).

Step2: For right - endpoints

The right - endpoints of the sub - intervals ([0,2],[2,4],[4,6],[6,8],[8,10]) are (x_1 = 2,x_2=4,x_3 = 6,x_4=8,x_5 = 10). Estimate the integral (\int_{0}^{10}f(x)dx\approx\sum_{i = 1}^{5}f(x_i)\Delta x=2\left[f(2)+f(4)+f(6)+f(8)+f(10)\right]). We estimate the function values from the graph: (f(2)\approx1,f(4)\approx0,f(6)\approx2,f(8)\approx - 1,f(10)\approx0). Then (2(1 + 0+2 - 1+0)=4).

Step3: For left - endpoints

The left - endpoints of the sub - intervals are (x_0 = 0,x_1=2,x_2 = 4,x_3=6,x_4 = 8). Estimate the integral (\int_{0}^{10}f(x)dx\approx\sum_{i = 0}^{4}f(x_i)\Delta x=2\left[f(0)+f(2)+f(4)+f(6)+f(8)\right]). From the graph, (f(0)\approx0,f(2)\approx1,f(4)\approx0,f(6)\approx2,f(8)\approx - 1). Then (2(0 + 1+0+2 - 1)=4).

Step4: For midpoints

The midpoints of the sub - intervals ([0,2],[2,4],[4,6],[6,8],[8,10]) are (x_1=1,x_2 = 3,x_3=5,x_4 = 7,x_5=9). Estimate the integral (\int_{0}^{10}f(x)dx\approx\sum_{i = 1}^{5}f(x_i)\Delta x=2\left[f(1)+f(3)+f(5)+f(7)+f(9)\right]). From the graph, (f(1)\approx1,f(3)\approx - 0.5,f(5)\approx1.5,f(7)\approx0,f(9)\approx - 0.5). Then (2(1-0.5 + 1.5+0 - 0.5)=2.5).

Answer:

(a) 4 (b) 4 (c) 2.5