Question

the graph of a function g is given. let f(x)=(int_{1}^{x}g(t)dt). (a) use the left riemann sum approximation with n = 3 subdivisions to approximate f(4). (b) use the left riemann sum approximation with n = 2 subdivisions to approximate f(3).

Explanation:

Step1: Recall Left - Riemann sum formula

The left - Riemann sum for $\int_{a}^{b}g(t)dt$ with $n$ sub - divisions is $L_n=\sum_{i = 0}^{n - 1}g(t_i)\Delta t$, where $\Delta t=\frac{b - a}{n}$ and $t_i=a + i\Delta t$.

Step2: Calculate for part (a)

For $f(4)=\int_{1}^{4}g(t)dt$ with $n = 3$ sub - divisions. Here, $a = 1$, $b = 4$, so $\Delta t=\frac{4 - 1}{3}=1$. The sub - intervals are $[1,2]$, $[2,3]$, $[3,4]$. The left - endpoints are $t_0 = 1$, $t_1 = 2$, $t_2 = 3$. From the graph, assume $g(1)\approx1$, $g(2)\approx2$, $g(3)\approx2$. Then $L_3=g(1)\times1+g(2)\times1+g(3)\times1=1\times1 + 2\times1+2\times1=5$.

Step3: Calculate for part (b)

For $f(3)=\int_{1}^{3}g(t)dt$ with $n = 2$ sub - divisions. Here, $a = 1$, $b = 3$, so $\Delta t=\frac{3 - 1}{2}=1$. The sub - intervals are $[1,2]$, $[2,3]$. The left - endpoints are $t_0 = 1$, $t_1 = 2$. From the graph, assume $g(1)\approx1$, $g(2)\approx2$. Then $L_2=g(1)\times1+g(2)\times1=1\times1+2\times1 = 3$.

Answer:

(a) 5
(b) 3