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Problem 27 Hard Difficulty

Use the Midpoint Rule with $ n = 5 $ to estimate the volume obtained by rotating about the y-axis the region under the curve $ y = \sqrt{1 + x^3} $ , $ 0 \le x \le 1 $.

Answer

$V \approx 3.680742$

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Video Transcript

remember the fact that the formula for volume is two pi times Ian to go from are lowered upper bound with 01 of radius times height, which in this case, is gonna be acts times one plus x cubed de ax. Remember the interval with is gonna be B minus over end. That's the formula. This is equivalent to plugging in one month, year over five, which is simply 1/5 which is 0.2 written as a decimal. Therefore, writing our chart acts and that our functions I'm gonna write this function is off our backs. We have 0.1 0.6 to 8, 0.3, 1.9, 1.9 ton. And then we have 0.53 point 33 0.75 point 09 0.97 point 43 Okay, now that we have this, we know we add up all these values on the right hand column. So 0.6 to 8 plus 1.9 10 plus 3.33 plus 5.97 point +43 multiplied by 0.2. We end up with 3.680742 Using this midpoint method