Final answer:
To estimate the volume of the solid formed by rotating the curve about the x-axis, we can use Simpson's rule. Simpson's rule involves dividing the interval into subintervals, calculating function values, and applying a weighted average formula. Finally, multiply the estimated area by the width of the interval to get the estimated volume.
Step-by-step explanation:
Simpson's rule is a numerical method used to estimate the area under a curve. To estimate the volume of the solid formed by rotating the curve about the x-axis, we can use Simpson's rule to estimate the area between the curve and the x-axis. Here's how:
- Divide the interval [0, 1] into an even number of subintervals. In this case, we have 5 subintervals.
- Calculate the width of each subinterval, which is the difference between consecutive x-values: h = (1-0)/5 = 0.2
- Calculate the function values at each x-coordinate using the given points.
- Apply Simpson's rule to estimate the area. I won't go into the details here, but the formula involves adding up a weighted average of the function values.
- Finally, multiply the estimated area by the width of the interval to get the estimated volume of the solid formed.