Question

approximate the area under the curve ( y = x^{2} ) from ( x = 1 ) to ( x = 4 ) using a right endpoint approximation with 6 subdivisions.

Explanation:

Step1: Calculate the width of each sub - interval

The formula for the width of a sub - interval \(\Delta x=\frac{b - a}{n}\), where \(a = 1\), \(b = 4\), and \(n=6\).
\(\Delta x=\frac{4 - 1}{6}=\frac{3}{6}=0.5\)

Step2: Find the right - endpoint \(x_i\)

The right - endpoints \(x_i=a + i\Delta x\), for \(i = 1,2,\cdots,6\).
When \(i = 1\), \(x_1=1+0.5\times1 = 1.5\)
When \(i = 2\), \(x_2=1+0.5\times2 = 2\)
When \(i = 3\), \(x_3=1+0.5\times3 = 2.5\)
When \(i = 4\), \(x_4=1+0.5\times4 = 3\)
When \(i = 5\), \(x_5=1+0.5\times5 = 3.5\)
When \(i = 6\), \(x_6=1+0.5\times6 = 4\)

Step3: Calculate the function values at the right - endpoints

\(y_1=(1.5)^2 = 2.25\), \(y_2=(2)^2 = 4\), \(y_3=(2.5)^2 = 6.25\), \(y_4=(3)^2 = 9\), \(y_5=(3.5)^2 = 12.25\), \(y_6=(4)^2 = 16\)

Step4: Use the Right - Endpoint formula \(R_n=\Delta x\sum_{i = 1}^{n}f(x_i)\)

\(R_6=0.5\times(2.25 + 4+6.25 + 9+12.25 + 16)\)
\(R_6=0.5\times(49.75)\)

Answer:

\(24.875\)