Find the volume of the solid formed by rotating the region bounded by the graph of y=1 + √x , the y-axis, and the line y = 3 about the x-axis.

Question

Find the volume of the solid formed by rotating the region bounded by the graph of  , the y-axis, and the line y = 3 about the x-axis

8.376

41.888

26.667

none of these

Answer 1

I am a bit perplexed! The area in question is the area bounded by y = 3 and the graph y = 1 + sqrt(x).

So it is the area from y= 1 + sqrt(x) on the bottom and y=3 on the top. Then this bounded area is to be revolved around the x axis to form a volume.

The bounded area is 2.6666, the y center of area (ybar) value is 2.5.

Then the volume generated by revolving this area around the x axis is given by the Pappus Formula

V = 2*pi*ybar*area = 2*pi*(2.5)*(2.6666) = 41.89 (Please Comment why this answer is incorrect!)

Answer 2

Draw a diagram.

Using the disk method, the volume of a typical disk is π[3 - (1+√x)]²dx

So, the volume of the solid is π∫_{(from 0 to 4)}(2 - √x)²dx

=π∫_{(from 0 to 4)}[4 - 4√x + x]dx

= π[4x - (8/3)x^3/2 + (1/2)x²] evaluated from 0 to 4

= π(16 - 64/3 + 8) ≈ 8.3776