Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{1}^{5} \frac{\sqrt{x-1}}{x} d x $$

Step-by-Step Solution

Answer

The above steps will help in approximating the value of the definite integral using the Trapezoidal Rule. Make sure to carry out all calculations correctly for an accurate approximation.

Step 1 - Calculate the width of each subinterval

First, find the width of each sub-interval by using the formula \(h = \frac{b-a}{n}\). Here, \(a = 1\), \(b = 5\), and \(n = 4\), so \(h = \frac{5-1}{4} = 1\). So, each sub-interval has a width of 1.

Step 2 - Create the sub-intervals

Now we need to determine the \(x\)-values that form the sub-intervals. Do this by starting at \(a\), and adding \(h\) until reaching \(b\). For this problem, the \(x\)-values will be \(1, 2, 3, 4, 5\). These are the input values at which we will evaluate the function.

Step 3 - Evaluate the function at each x-value

Now, calculate the function value at each x-value in the sub-intervals. The function is \(f(x) = \frac{\sqrt{x-1}}{x}\), so calculate \(f(1), f(2), f(3), f(4)\), and \(f(5)\). Remember to handle f(1) carefully, since \(x - 1 = 0\) in the square root function.

Step 4 - Apply the trapezoid rule formula

Now, let's approximate the integral using the trapezoid rule. The formula for the trapezoid rule is: \(\int_{a}^{b} f(x )dx \approx \frac{h}{2}[f(x_{0})+2f(x_{1})+2f(x_{2})+...+2f(x_{n-1})+f(x_{n})]\). Here, \(x_{0} = 1\), \(x_{1} = 2\), \(x_{2} = 3\), \(x_{3} = 4\), and \(x_{4} = 5\). Plug the values of \(f(x_{i})\) and \(h\) into the formula to get the approximate integral value.