Use the Midpoint Rule with n = 5 to estimate the volume V obtained by rotating about the y-axis the region under the curve y = 4 + 6x3, 0 ≤ x ≤ 1. (Round your answer to two decimal places.)V =

To solve this problem, we need to use the formula for the volume of a solid of revolution using the Midpoint Rule. The formula is:

V = π ∑ [f(xi)]² Δx

where f(xi) is the function value at the midpoint of each interval and Δx is the width of each interval.

Step 1: Determine the width of each interval (Δx) Since we are given that n = 5 and the interval of x is from 0 to 1, we can calculate Δx as (1-0)/5 = 0.2.

Step 2: Determine the midpoints of each interval (xi) The midpoints of each interval are 0.1, 0.3, 0.5, 0.7, and 0.9.

Step 3: Evaluate the function at each midpoint and square the result We substitute each xi into the function y = 4 + 6x³ and square the result.

f(0.1)² = (4 + 6*(0.1)³)² = 16.1464 f(0.3)² = (4 + 6*(0.3)³)² = 18.9744 f(0.5)² = (4 + 6*(0.5)³)² = 24.01 f(0.7)² = (4 + 6*(0.7)³)² = 30.6601 f(0.9)² = (4 + 6*(0.9)³)² = 39.2209

Step 4: Sum up the results and multiply by π and Δx V = π * Δx * (f(0.1)² + f(0.3)² + f(0.5)² + f(0.7)² + f(0.9)²) V = π * 0.2 * (16.1464 + 18.9744 + 24.01 + 30.6601 + 39.2209) V = π * 0.2 * 128.0118 V = 25.60π

So, the volume V obtained by rotating the region under the curve y = 4 + 6x³ about the y-axis from x = 0 to x = 1 is approximately 80.42 cubic units (rounded to two decimal places).