4) approximate the area under the following curve (from x = 1 to x = 17) using left and right hand riemann…

Explanation:

Step1: Determine the number of sub - intervals

The interval is from $x = 1$ to $x=17$ with $\Delta t = 2$. The number of sub - intervals $n=\frac{17 - 1}{2}=8$. The endpoints of the sub - intervals are $x_0 = 1,x_1=3,x_2 = 5,\cdots,x_8=17$.

Step2: Calculate the left - hand Riemann sum

The left - hand Riemann sum $L=\sum_{i = 0}^{7}f(x_i)\Delta t$. We need to estimate the function values $f(x_i)$ from the graph at $x_0 = 1,x_1=3,x_2 = 5,x_3=7,x_4 = 9,x_5=11,x_6 = 13,x_7=15$. Let's assume the function values are $f(1)\approx0,f(3)\approx4,f(5)\approx8,f(7)\approx10,f(9)\approx10,f(11)\approx8,f(13)\approx4,f(15)\approx0$. Then $L=(0 + 4+8 + 10+10+8+4+0)\times2=(44)\times2 = 88$.

Step3: Calculate the right - hand Riemann sum

The right - hand Riemann sum $R=\sum_{i = 1}^{8}f(x_i)\Delta t$. The function values are estimated at $x_1=3,x_2 = 5,x_3=7,x_4 = 9,x_5=11,x_6 = 13,x_7=15,x_8=17$. Assuming $f(3)\approx4,f(5)\approx8,f(7)\approx10,f(9)\approx10,f(11)\approx8,f(13)\approx4,f(15)\approx0,f(17)\approx0$. Then $R=(4 + 8+10 + 10+8+4+0+0)\times2=(44)\times2=88$.

Answer:

Left - hand Riemann sum: 88, Right - hand Riemann sum: 88