Find the volume of the solid formed by revolving the region bounded by latex: y = \sqrt{x} y = x and the lines latex: y = 1 y = 1 and latex: x = 4 x = 4 about the line latex: y = 1 y = 1 .

The volume is:

Step-by-step explanation:

See the sketch of the region in the attached graph.

We set the integral using washer method:

Notice here the radius of the washer is the difference of the given curves:

So the integral becomes:

We solve it:

Factor out and distribute the exponent (you can use FOIL):

Notice:

So the integral becomes:

Then using the basic rule to evaluate the integral:

Simplifying a bit:

Then plugging the limits of the integral:

Taking the root (rational exponents):

Then doing those arithmetic computations we get:

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note: the best i know.

1) 2/5

2) zero=0/1

3) zero=0/2

4) -4/5

5) undefined=3/0