James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

discovery project Rotating on a Slant 557
discovery Project
Rotating on a Slant
We know how to find the volume of a solid of revolution obtained by rotating a region about a
horizontal or vertical line (see Section 6.2). We also know how to find the surface area of a surface
of revolution if we rotate a curve about a horizontal or vertical line (see Section 8.2). But
what if we rotate about a slanted line, that is, a line that is neither horizontal nor vertical? In this
project you are asked to discover formulas for the volume of a solid of revolution and for the area
of a surface of revolution when the axis of rotation is a slanted line.
Let C be the arc of the curve y − f sxd between the points Psp, f spdd and Qsq, f sqdd and let 5
be the region bounded by C, by the line y − mx 1 b (which lies entirely below C), and by the
perpendiculars to the line from P and Q.
y
y=ƒ
Q
P
C
y=mx+b
Îu
0 p
q
x
1. Show that the area of 5 is
1
y q
f f sxd 2 mx 2 bg f1 1 mf 9sxdg dx
1 1 m 2 p
[Hint: This formula can be verified by subtracting areas, but it will be helpful throughout the
project to derive it by first approximating the area using rectangles perpendicular to the line,
as shown in the following figure. Use the figure to help express Du in terms of Dx.]
tangent to C
at {x i , f(x i )}
?
?
y=mx+b
Îu
å
x i
∫
Îx
y
(2π, 2π)
2. Find the area of the region shown in the figure at the left.
y=x+sin x
y=x-2
3. Find a formula (similar to the one in Problem 1) for the volume of the solid obtained by
rotating 5 about the line y − mx 1 b.
4. Find the volume of the solid obtained by rotating the region of Problem 2 about the
line y − x 2 2.
0
x
CAS
5. Find a formula for the area of the surface obtained by rotating C about the line y − mx 1 b.
6. Use a computer algebra system to find the exact area of the surface obtained by rotating the
curve y − sx , 0 < x < 4, about the line y − 1 2 x. Then approximate your result to three
decimal places.
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