4. TANGENT PLANES AND LINEAR APPROXIMATIONS
83
Example. Consider the solid obtained when rotating the following region
about the x-axis. Note that the region is composed of a right triangle and a
quarter-circle.
(1) Compute the volume V (x, y) of this solid.
1 2
1 ⇣ 4 3⌘
1
14
28
V = ⇡r h +
⇡r
= ⇡(22)(3) + ⇡(23) = ⇡ ⇡ 29.3215
3
|3 {z } |2 3{z }
|3
{z 2 3
}
cone hemisphere
specific case
|
{z
}
general formula
(2)Find the volume V (x, y) of a similar solid created by rotating a region with
horizontal dimension x and vertical dimension y instead of 3 and 2.
1
2
V = ⇡x2y + ⇡x3
3
3
(3) Oh yeah — we forgot to tell you in (1) that the “2” and the “3” were really
just rounded-o↵ numbers. The actual quantities can be o↵ by up to 0.5 in each
direction. Use linear approximation to estimate the maximum possible error in
your answer to (1).
|dx|  0.5,
|dy|  0.5
dV (x, y) = Vx(x, y)dx + Vy (x, y)dy
2
2
Vx(x, y) = ⇡xy + 2⇡x2 =) Vx(3, 2) = ⇡(3)(2) + 2⇡(32) = 22⇡ ⇡ 69.1150
3
3
1
1
Vy (x, y) = ⇡x2 =) Vy (3, 2) = ⇡(32) = 3⇡ ⇡ 9.4248
3
3
|dV (3, 2)|  69.1150(.5) + 9.4248(.5) = 39.2699