Please Help NO LINKS Suppose that
R
is the finite region bounded by
f
(
x
)
=
4
√
x
and
g
(
x
)
=
x
.

Find the exact value of the volume of the object we obtain when rotating
R
about the
x
-axis.

V
=

Find the exact value of the volume of the object we obtain when rotating
R
about the
y
-axis.

V
=

Part A)

2048π/3 cubic units.

Part B)

8192π/15 units.

Step-by-step explanation:

We are given that R is the finite region bounded by the graphs of functions:

Part A)

We want to find the volume of the solid of revolution obtained when rotating R about the x-axis.

We can use the washer method, given by:

Where R is the outer radius and r is the inner radius.

Find the points of intersection of the two graphs:

Hence, our limits of integration is from x = 0 to x = 16.

Since 4√x ≥ x for all values of x between [0, 16], the outer radius R is f(x) and the inner radius r is g(x). Substitute:

Evaluate:

The volume is 2048π/3 cubic units.

Part B)

We want to find the volume of the solid of revolution obtained when rotating R about the y-axis.

First, rewrite each function in terms of y:

Solving for the intersection yields y = 0 and y = 16. So, our limits of integration are from y = 0 to y = 16.

The washer method for revolving about the y-axis is given by:

Since g(y) ≥ f(y) for all y in the interval [0, 16], our outer radius R is g(y) and our inner radius r is f(y). Substitute and evaluate:

The volume is 8192π/15 cubic units.

2.83941151 would be the answer you're looking for! : )

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step-by-step explanation: