CK-12-Calculus

5.2. Volumes http://www.ck12.org

(^12) b
(^12) a=
h−y
h ,
b=ah(h−y).
Since the cross-sectional area atyisA(y) =b^2 ,
A(y) =b^2 =a
2
h^2 (h−y)
(^2).
Using the volume formula,

V=

∫d

c A(y)dy

=

∫h

0

a^2

h^2 (h−y)

(^2) dy
=a
2
h^2
∫h
0 (h−y)
(^2) dy.
Usingu−substitution to integrate, we eventually get
V=a
2
h^2

[

−^13 (h−y)^3

]h

0

=^13 a^2 h.

Therefore the volume of the pyramid isV=^13 a^2 h, which agrees with the standard formula.

Volumes of Solids of Revolution

The Method of Disks
Suppose a functionfis continuous and non-negative on the interval[a,b],and suppose thatRis the region between
the curvefand thex−axis (Figure 8a). If this region is revolved about thex−axis, it will generate a solid that will
have circular cross-sections (Figure 8b) with radii off(x)at eachx.