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:

equal to

step-by-step explanation:

b) k’(6, 4); l’(3, 3)

step-by-step explanation:

a 270° clockwise rotation maps every point (x, y) to (y, -x). (it negates the x-coordinate and then switches it and the y-coordinate.)

this means the endpoints of the side congruent to kl, k' and l', would be mapped as:

k'(-4, 6)→(6, 4)

l'(-3, 3)→(3, 3)