Question

1. use ( n = 3 ) subintervals and left - endpoints to approximate ( int_{0}^{6} h(x) dx ).

draw the three rectangles on the graph.

Explanation:

Step1: Calculate the width of sub - intervals

The formula for the width of sub - intervals is $\Delta x=\frac{b - a}{n}$. Here, $a = 0$, $b = 6$, and $n = 3$. So, $\Delta x=\frac{6-0}{3}=2$.

Step2: Determine the left - endpoints

The left - endpoints of the sub - intervals are $x_0=a = 0$, $x_1=a+\Delta x=0 + 2=2$, $x_2=a + 2\Delta x=0+2\times2 = 4$.

Step3: Estimate the function values at left - endpoints

From the graph:

• When $x = 0$, $h(0)=2$
• When $x = 2$, $h(2)=0$
• When $x = 4$, $h(4)=-4$

Step4: Apply the left - endpoint Riemann sum formula

The left - endpoint Riemann sum formula is $L_n=\sum_{i = 0}^{n-1}h(x_i)\Delta x$.
Substitute $n = 3$, $\Delta x=2$, $h(x_0)=2$, $h(x_1)=0$, $h(x_2)=-4$ into the formula:
$L_3=h(0)\times\Delta x+h(2)\times\Delta x+h(4)\times\Delta x$
$L_3=(2)\times2+(0)\times2+(-4)\times2$
$L_3=4 + 0-8$

Answer:

$-4$