Work and the Average Value of a Function

Two more applications of the "slice, approximate, integrate" principle: work and average value. Work integrates a variable force over a displacement — the interest here is less the physics and more the fact that the formula $W = \int_a^b F(x)\,dx$ is structurally identical to the present-value integral $\int_0^T r \cdot B(t)\,dt$ (the accumulated interest paid by a bond) and to any path integral in finance. The Average Value Theorem ($f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$) directly computes time-averaged portfolio returns, average interest rates over a period, and expected asset prices under a uniform-time model.

Learning objectives. By the end of this note you should be able to: