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

Explanation:

Step1: Determine sub - interval width

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

Step2: (a) Right - endpoints

The right - endpoints of the sub - intervals $[0,2],[2,4],[4,6],[6,8],[8,10]$ are $x_1 = 2,x_2 = 4,x_3 = 6,x_4 = 8,x_5 = 10$. We estimate the integral using $R_5=\sum_{i = 1}^{5}f(x_i)\Delta x$. By looking at the graph and estimating the function values at these points and multiplying by $\Delta x = 2$, we sum up the products.

Step3: (b) Left - endpoints

The left - endpoints of the sub - intervals $[0,2],[2,4],[4,6],[6,8],[8,10]$ are $x_0 = 0,x_1 = 2,x_2 = 4,x_3 = 6,x_4 = 8$. We estimate the integral using $L_5=\sum_{i = 0}^{4}f(x_i)\Delta x$. By looking at the graph and estimating the function values at these points and multiplying by $\Delta x = 2$, we sum up the products.

Step4: (c) Midpoints

The midpoints of the sub - intervals $[0,2],[2,4],[4,6],[6,8],[8,10]$ are $x_{0.5}=1,x_{1.5}=3,x_{2.5}=5,x_{3.5}=7,x_{4.5}=9$. We estimate the integral using $M_5=\sum_{i = 0}^{4}f(x_{i + 0.5})\Delta x$. Since we know $M_5 = 6$ (given as correct in the problem).

Answer:

(a) [Estimated value from graph using right - endpoints] (b) [Estimated value from graph using left - endpoints] (c) 6