| dA2 = - | 12 l² | y²dA + 9dA, |
and
| A2 = - | 12 l² | ∫y²dA + 9A. |
But the integral gives the moment of inertia I of the area A about the axis XX. As A2 is proportional to the roll of W2, A to that of W, we can write
I = Cw - C2 w2,
Ay = C1 w1,
A = Cc w.
If a line be drawn parallel to the axis XX at the distance y, it will pass through the mass-centre of the given figure. If this represents the section of a beam subject to bending, this line gives for a proper choice of XX the neutral fibre. The moment of inertia for it will be I + Ay². Thus the instrument gives at once all those quantities which are required for calculating the strength of the beam under bending. One chief use of this integrator is for the calculation of the displacement and stability of a ship from the drawings of a number of sections. It will be noticed that the length of the figure in the direction of XX is only limited by the length of the rail.
This integrator is also made in a simplified form without the wheel W2. It then gives the area and first moment of any figure.
While an integrator determines the value of a definite integral, hence a Integraphs. mere constant, an integraph gives the value of an indefinite integral, which is a function of x. Analytically if y is a given function f(x) of x and