Question

estimate the area under the curve ( f(x) = 2sqrt{x - 1} ) from ( x = 1 ) to ( x = 17 ) by using three trapezoids over the sub-intervals 1, 5, 5, 10, and 10, 17.

Explanation:

Step1: Recall Trapezoid Area Formula

The area of a trapezoid with bases \(b_1\), \(b_2\) and height \(h\) is \(A=\frac{1}{2}(b_1 + b_2)h\). For each sub - interval \([x_i,x_{i + 1}]\), \(b_1=f(x_i)\), \(b_2 = f(x_{i+1})\) and \(h=x_{i + 1}-x_i\).

Step2: Calculate for Sub - interval \([1,5]\)

• First, find \(f(1)\) and \(f(5)\):
• For \(x = 1\), \(f(1)=2\sqrt{1 - 1}=2\times0 = 0\).
• For \(x = 5\), \(f(5)=2\sqrt{5 - 1}=2\times2 = 4\).
• The width of the sub - interval \(h_1=5 - 1=4\).
• The area of the trapezoid for \([1,5]\) is \(A_1=\frac{1}{2}(f(1)+f(5))\times(5 - 1)=\frac{1}{2}(0 + 4)\times4=\frac{1}{2}\times4\times4 = 8\).

Step3: Calculate for Sub - interval \([5,10]\)

• Find \(f(5)\) and \(f(10)\):
• We already know \(f(5) = 4\).
• For \(x = 10\), \(f(10)=2\sqrt{10 - 1}=2\times3 = 6\).
• The width of the sub - interval \(h_2=10 - 5 = 5\).
• The area of the trapezoid for \([5,10]\) is \(A_2=\frac{1}{2}(f(5)+f(10))\times(10 - 5)=\frac{1}{2}(4 + 6)\times5=\frac{1}{2}\times10\times5=25\).

Step4: Calculate for Sub - interval \([10,17]\)

• Find \(f(10)\) and \(f(17)\):
• We know \(f(10)=6\).
• For \(x = 17\), \(f(17)=2\sqrt{17 - 1}=2\times4 = 8\).
• The width of the sub - interval \(h_3=17 - 10 = 7\).
• The area of the trapezoid for \([10,17]\) is \(A_3=\frac{1}{2}(f(10)+f(17))\times(17 - 10)=\frac{1}{2}(6 + 8)\times7=\frac{1}{2}\times14\times7 = 49\).

Step5: Total Area

The total area \(A=A_1+A_2+A_3\).
Substitute the values of \(A_1 = 8\), \(A_2=25\) and \(A_3 = 49\) into the formula:
\(A=8 + 25+49=82\).

Answer:

The estimated area under the curve is \(82\).