Simpson's Rule.-The base line must be divided into an The area is then equal to

even number of equal parts.

in which a+n is the sum of the end ordinates; 42h is four times the sum of all intermediate even-numbered ordinates;

ple, what is the area ABCD according to Simpson's rule? SOLUTION.— A=[19+23+4(18+12+17)+2(14+13)]× =4,733 sq. li.

AREA BOUNDED BY AN IRREGULAR CURVE Suppose that it is required to find the area enclosed by the heavy irregular curve shown in the accompanying illustration. F

MN

H

A broken line AEFMGHIA is drawn around the curved boundary line and as close to it as convenient. Ordinates to the straight lines thus drawn are measured from the points

where the direction of the curved boundary changes materially. as shown. The area of the polygon AEFMGHIA is calculated by one of the methods previously explained, and from it is subtracted the sum of the areas included between the curved boundary and the broken line, calculated in the manner just shown.

At such corners as A, the triangles ABC and ABD are computed from the measured bases AC and AD and the altitudes BC and BD. All the quadrilaterals, as QRST, are treated as trapezoids; and such three-sided figures as MPN, as triangles.

*The perimeter of an ellipse cannot be exactly determined without a very elaborate calculation, and this formula is merely an approximation giving fairly close results.

A prismoid is a solid having two parallel plane ends, the edges of which are connected by plane triangular or quadrilateral surfaces. = area of one end;

A

a area of other end;

m = area of section midway between ends;

l=perpendicular distance between ends.

V=l(A+a+4m)

The area m is not in general a mean between the areas of the two ends, but its sides are means between the corresponding lengths of the ends.

To obtain area of base, divide it into triangles, and find the sum of their areas.

The formula for V applies to any pyramid whose base is A and altitude h.

FRUSTUM OF REGULAR PYRAMID

a= area of upper base;

A = area of lower base;
p-perimeter of upper base;
P= perimeter of lower base.
C = (P+b)

S=(P+b)+A+a
V=th(A+a+ VAα)

The formula for V applies to the frustum of any pyramid.

For prisms with regular polygons as

bases, P= length of one side X number of sides.

To obtain area of base, if it is a polygon, divide it into tri

angles, and find sum of partial areas.