Final answer:
The Midpoint Rule is used to approximate definite integrals using rectangles. To use this rule, we divide the interval of integration into subintervals of equal width. We then evaluate the function at the midpoints of each subinterval and multiply by the width of each subinterval.
Step-by-step explanation:
The Midpoint Rule is a method of approximating definite integrals using rectangles. In this case, we are asked to use the Midpoint Rule to approximate the integral of sin(sqrt(x)) dx. To use the Midpoint Rule, we need to divide the interval of integration into n subintervals of equal width. The width of each subinterval is given by (b-a)/n, where a and b are the lower and upper limits of integration. Let's assume n = 4 for this example.
Step 1: Calculate the width of each subinterval: (b-a)/n = (1-0)/4 = 1/4 = 0.25
Step 2: Determine the midpoints of each subinterval. The midpoint of the first subinterval is 0.25/2 = 0.125, the midpoint of the second subinterval is 0.375, the midpoint of the third subinterval is 0.625, and the midpoint of the fourth subinterval is 0.875.
Step 3: Evaluate the function at each midpoint and multiply by the width of each subinterval. In this case, we evaluate sin(sqrt(x)) at each midpoint: sin(sqrt(0.125))*0.25 + sin(sqrt(0.375))*0.25 + sin(sqrt(0.625))*0.25 + sin(sqrt(0.875))*0.25.
Calculating these values, we get approximately 0.0694. Therefore, the integral of sin(sqrt(x)) dx using the Midpoint Rule with n=4 is approximately 0.0694.