Question

approximate the area under the curve graphed below from ( x = 2 ) to ( x = 7 ) using a left endpoint approximation with 5 subdivisions.

Explanation:

Step1: Calculate the width of each subdivision

The interval is from \(x = 2\) to \(x=7\). The formula for the width \(\Delta x=\frac{b - a}{n}\), where \(a = 2\), \(b = 7\), and \(n = 5\).
\(\Delta x=\frac{7 - 2}{5}=1\)

Step2: Determine the left - endpoint values

The left - endpoints of the sub - intervals are \(x_0=2\), \(x_1 = 3\), \(x_2=4\), \(x_3=5\), \(x_4=6\)
From the graph:
When \(x = 2\), \(y_0=2\)
When \(x = 3\), \(y_1=2.5\)
When \(x = 4\), \(y_2=2.8\)
When \(x = 5\), \(y_3=3\)
When \(x = 6\), \(y_4=3.2\)

Step3: Apply the Left - Endpoint formula

The Left - Endpoint formula is \(L_n=\sum_{i = 0}^{n-1}f(x_i)\Delta x\)
Since \(\Delta x = 1\), \(L_5=(y_0 + y_1+y_2+y_3+y_4)\times\Delta x\)
\(L_5=(2 + 2.5+2.8+3+3.2)\times1\)
\(L_5=13.5\)

Answer:

\(13.5\)