Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{0}^{4} \sqrt{1+x^{2}} d x $$

Step-by-Step Solution

Answer

The definite integral is approximated as \(1/2 + \sqrt{2} + \sqrt{5} + \sqrt{10} + \sqrt{17}/2\).

Step 1: Calculate the width of the subintervals

The width of each subinterval, denoted as \(h\), can be found by dividing the interval width by the number of subintervals: \(h = (b-a)/n = (4-0)/4 = 1\). Therefore, the width of each subinterval is 1.

Step 2: Calculate the function values at each point

Next, compute the function's values at the endpoints of the subintervals: \(f(0)\), \(f(1)\), \(f(2)\), \(f(3)\), and \(f(4)\). Using the function \(f(x) = \sqrt{1+x^2}\), we find: \(f(0) = \sqrt{1+0^2} = 1\), \(f(1) = \sqrt{1+1^2} = \sqrt{2}\), \(f(2) = \sqrt{1+2^2} = \sqrt{5}\), \(f(3) = \sqrt{1+3^2} = \sqrt{10}\), \(f(4) = \sqrt{1+4^2} = \sqrt{17}\).

Step 3: Insert the values into the Trapezoidal Rule formula

According to the Trapezoidal rule, the definite integral can be approximated as: \(\int_{a}^{b}f(x) dx \approx (h/2)*[f(a) + 2*f(x_{1}) + 2*f(x_{2}) + ... + 2*f(x_{n-1}) + f(b)]\). So the approximate integral is: \((1/2)*[1+2*\sqrt{2}+2*\sqrt{5}+2*\sqrt{10}+\sqrt{17}]\).

Step 4: Simplify to find the definite integral

Simplify the expression to find the final value for the definite integral. The final result is: \(1/2 + \sqrt{2} + \sqrt{5} + \sqrt{10} + \sqrt{17}/2\). This is the approximate value of the given integral using the Trapezoidal Rule with \(n=4\).