INTEGRALS FOR SURFACE AREA, WORK, CENTROIDS
257
31.26
The region bounded by y = sin x, y = 0, from x = 0 to x = IT (Fig. 31-4).
By symmetry,
The area
The moment about the
By Problem 29.40,
Hence,
Thus,
Fig. 31-4
Fig. 31-5
31.27
The region bounded by y =
(Fig. 31-5).
y=0, * = 1, x = 2
The area
Thus,
The moment about the *-axis is
The moment about the y-axis is
So the centroid is
Thus,
31.28
A right triangle with legs r and h.
Let the rieht ansle be at the origin, and let the legs r and h be along the positive x-axis and y-axis, respectively
(Fig. 31-6). The hypotenuse is along the line
The area A is
The moment about the
Hence,
In
y-axis is
similar manner,
Fig. 31-6
Fig. 31-7
31.29
The region bounded by y = x
2
and y=jt (Fig. 31-7).
The area
The moment about y-axis is
The moment about the x-axis
Thus,
Thus,
31.30
The region bounded by y = e", y = e ', and x = 1 (Fig. 31-8).
*-axis