QUESTION 1
Approximate $\int_{0}^{2} \sqrt{1+x^{4}} d x$ by using Reimann Sums with right end point rectangles with 4 rectangles. Round your answer to three digits after the decimal sign.

Step 1 ：We are given the function \(f(x) = \sqrt{1+x^{4}}\) and we are asked to approximate the definite integral of this function from 0 to 2 using Riemann Sums with right end point rectangles and 4 rectangles.

Step 2 ：The formula for the Riemann Sum is \(\sum_{i=1}^{n} f(x_i) \Delta x\), where \(f(x_i)\) is the value of the function at \(x_i\), \(\Delta x\) is the width of each rectangle, which is equal to \((b - a) / n\), where \(a\) and \(b\) are the limits of integration and \(n\) is the number of rectangles.

Step 3 ：In this case, \(a = 0\), \(b = 2\), \(n = 4\), and \(f(x) = \sqrt{1+x^{4}}\). We will use right end point rectangles, which means that the height of each rectangle is determined by the value of the function at the right end point of the rectangle.

Step 4 ：First, we calculate \(\Delta x = (b - a) / n = (2 - 0) / 4 = 0.5\).

Step 5 ：Next, we calculate the x-values for the right end points of the rectangles. These are \(0.5, 1, 1.5,\) and \(2\).

Step 6 ：We then calculate the value of the function at these x-values and multiply by \(\Delta x\) to get the area of each rectangle. Summing these areas gives the Riemann Sum.

Step 7 ：The Riemann Sum is approximately 4.515155022422098.

Step 8 ：Rounding this to three decimal places gives \(\boxed{4.515}\).