Question

Use the midpoint rule with the given value of n to approximate the integral. Round the answer to four decimal places. ∫(3x sin²(x)) dx, 0 to n = 4

Answer 1

Using the midpoint rule with n=4, we divide the interval [0, π] into four equal subintervals and use the midpoint of each subinterval to approximate the integral of 3x sin²(x) dx, with the result rounded to four decimal places.

Step-by-step explanation:

The question asks us to use the midpoint rule to approximate the integral ∫(3x sin²(x)) dx from 0 to π (where 'n' is a stand-in for π) with n = 4 subintervals. To do this, we will divide the interval [0, π] into 4 equal subintervals and use the midpoint of each subinterval to evaluate the integrand function. Then, we simply multiply each evaluation by the width of the subintervals and sum them up to get the approximation. The approximation is as follows:

• Subinterval width: ∆x = π/4
• Midpoint of each subinterval: x1 = π/8, x2 = 3π/8, x3 = 5π/8, x4 = 7π/8
• Evaluation of the function at each midpoint: f(x1), f(x2), f(x3), and f(x4)
• Approximated integral: ∫(3x sin²(x)) dx ≈ ∆x * [f(x1) + f(x2) + f(x3) + f(x4)]

After calculating, the result will be rounded to four decimal places as required.