The definite integral of f(x) is the difference between two values of the integral of f(x) for two distinct values of the variable x. If the integral of f(x) dx = F(x) + C, the definite integral is denoted by the symbol
 

The quantity F(b) - F(a) is called the definite integral of f(x) between the limits a and b or simply the definite integral from a to b. It is called the definite integral because the result involves neither x nor the constant C and therefore has a definite value. The numbers a and b are called the limits of integration, a being the lower limit and b the upper limit.
 

Integration Formulas

In these formulas, u and v denote differentiable functions of some independent variable (say x) and a, n, and C are constants.
 

1. The integral of the differential of a function u is u plus an arbitrary constant C (the definition of an integral).
    

   $$\displaystyle \int du = u + C$$
    

Indefinite Integrals

If F(x) is a function whose derivative F'(x) = f(x) on certain interval of the x-axis, then F(x) is called the anti-derivative of indefinite integral f(x). When we integrate the differential of a function we get that function plus an arbitrary constant. In symbols we write
 

Problem 648
For the cantilever beam loaded as shown in Fig. P-648, determine the deflection at a distance x from the support.
 

Problem 647
Find the maximum value of EIδ for the beam shown in Fig. P-647.
 

Problem 646
For the beam shown in Fig. P-646, determine the value of I that will limit the maximum deflection to 0.50 in. Assume that E = 1.5 × 10⁶ psi.
 

Problem 645
Compute the deflection and slope at a section 3 m from the wall for the beam shown in Fig. P-645. Assume that E = 10 GPa and I = 30 × 10⁶ mm⁴.
 

Problem 644
Determine the maximum deflection for the beam loaded as shown in Fig. P-644.
 

Problem 643
Find the maximum value of EIδ for the cantilever beam shown in Fig. P-643.
 

Problem 642
Find the maximum deflection for the cantilever beam loaded as shown in Figure P-642 if the cross section is 50 mm wide by 150 mm high. Use E = 69 GPa.