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W04 - HW Solutions

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02 - Surface area of revolved cubic

The curve over is revolved around the -axis. Find the area of the resulting surface.

Solution

Write out formula for surface area of surface revolved around -axis.

Plug in , , and .

Perform substitution: ,

Adjust bounds

Evaluate integral.

03 - Arc length of a curve - tricky algebra

Find the arc length of the curve for .

(Hint: expand under the root, then simplify, then factor; now it’s a square and the root disappears)

Solution

Write down formula for arc length of a curve.

Calculate

Plug in values to compute integral.

Compute integral.

04 - Arc length of a curve - tricky integration

Find the arc length of the curve for .

(Hint: the integral can be done using either: (i) -sub then trig sub, or (ii) ‘rationalization’ then partial fractions.)

Solution

Write down formula for arc length.

Set up integral.

.

Perform -substitution: ,

Adjust bounds:

Perform substitution:

Adjust bounds:

Compute partial fraction of integrand.

Expand:

Write down general PFD:

Solve for and :

Evaluate integral.

05 - Surface area of a cone

A cone may be described as the surface of revolution of a ray emanating from the origin, revolved around the -axis.

Let for some . Find the surface area of the cone given by revolving the graph of over .

Can you also calculate this area using geometry? And verify the two methods give the same formula? (Hint: ‘unroll’ the cone into a sector.)

Solution

Write out formula for surface area.

Calculate .

Set up integral by plugging in .

Evaluate integral. Note solution is just lateral area.

Verify with geometry. Note that unrolling the cone forms a sector with radius and arc length .

Radius is the lateral height of the cone: hypotenuse of right triangle with other sides and .

Arc length is , since is radius of the base.

Calculate area of sector using ratios.

Circumference of full circle:

Area of full circle:

06 - Surface area of a parabolic reflector

A parabolic reflector is given by rotating the curve around the y-axis for .

What is the surface area of this reflector?

Write out formula for surface area.

This problem can be done with respect to or with respect to . Note this will not always be the case.

Rewrite in terms of , and adjust bounds.

Compute and

Evaluate integrals.

with respect to : Note that since that is the radius in this case.

-substitution: , , .

with respect to :

-substitution: ,

07 - Surface area of torus

A torus is created by revolving about the x-axis the circle with this equation:

Find the surface area of this torus.

(Hint: compute for the top and bottom of the circle separately and add the results.)

Solution

Write down formula for surface area for surface revolved around -axis.

Identify formulas for bottom and top of the circle.

Let and

Compute and .

Compute integral using

Use trig substitution: .

Adjust bounds: