Definite Integrals

Engineering Context

Integrals and the definite definition of integrals is one of the basic principles of engineering as they are used to calculate varia areas, volumes, and masses across all disciplines of engineering.

  • MAE: Mechanical engineers may be tasked with designing and implementing pipes and channels for a system. The definite integral is very important in calculating the pressure, velocity, and volume flow rate of fluids in these pipes and channels. These kind of designs can be used in plans for pumps, turbines, and other fluid handling systems.
  • ECE: Electrical and Computer Engineers use definite integrals to calculate the energy, power, and properties of signals over time. These calculations are crucial to the designing of communication systems, control systems, and other systems that transmit and process signals.
  • BE: In the development of prosthetic devices, a biological engineer will use the definite integral to calculate the different forces acting on systems such as the joints and muscles while they are moving.
  • CEE: The definite integral is important to civil and environmental engineers in the building of structures such as beams and bridges that will withstand forces of human use (civil) as well as nature (environmental). The definite integral is used to calculate forces and stress of these structures under loads.

The Essentials

∫ a b f ( x ) d x = lim n → ∞ ∑ i = 1 n f ( x i * ) Δ x

Given an equation f that is continuous on an interval from a to b, the definite integral is the area under the curve f within the interval [a, b]. Compared with the integral of a function (called an indefinite integral to contrast it with a definite integral), a definite integral looks like this:

(a) Indefinite Integral

(a) Indefinite Integral

(b) Definite Integral

(b) Definite Integral

Evaluating an indefinite integral will yield a function representing the entire area under the curve. A definite integral will yield a numerical value.

A Deeper Dive

If f ( x ) is continuous on an interval [a, b], the definite integral of f from a to b is given by ∫ a b f ( x ) d x = lim n → ∞ ∑ i = 1 n f ( x i ) Δ x provided that the limit exists. In practice, the definite integral is usually evaluated by first evaluating the indefinite integral:

\[ \int_{a}^{b} f(x) \, dx = F(x) \bigg|_{a}^{b} \]

The function F(x) is then evaluated at a and b, and the difference is calculated to find the value of the definite integral:

\[ F(x) \bigg|_{a}^{b} = F(b) - F(a) \]

Consider the first example on this page. The value of the equation is f(x) = x2, and we will find the definite integral over the interval [3, 6]. First, we find the indefinite integral:

\[ \int_{a}^{b} f(x) \, dx = \int_{3}^{6} x^2 \, dx = \frac{1}{3} x^3 \bigg|_{3}^{6} \]
(a) Indefinite Integral

We then evaluate the value of the integral from a to b:

\[ \frac{1}{3} x^3 \bigg|_{3}^{6} = \frac{1}{3} 6^3 - \frac{1}{3} 3^3 = 72 - 9 = 63 \]

When evaluating an indefinite integral, we add a constant of integration C because part of the original function is lost when we calculate the derivative. There is not enough information to determine the original function from its derivative alone

Definite Integral of f(x)

However, definite integrals do not need to account for the constant of integration because it is canceled out. Since the constant of integration is constant [citation needed], the C value in F(b) is the same as the C in F(a). When you evaluate F(b) - F(a), the Cs cancel out.

Practice

Evaluate the definite integral ∫ a b f ( x ) :

  1. f ( x ) = x 2 ,   a = 0 ,   b = 7
  2. f ( x ) = sin ⁡ ( x ) ,   a = π / 2 ,   b = − π / 2
  3. \( f(x) = e^{(2x)}, \quad a = 0, \quad b = \ln 5 \)

The kinematic equations model the relationships of time, displacement, velocity, and acceleration. The function that describes acceleration is the derivative of the function for velocity (\( a = \frac{dv}{dt} \)). The function that describes velocity is the derivative of the function for displacement (\(v = \frac{dv}{dt}\)). Using this information and your knowledge of definite integrals, derive the following kinematic equations (t = time, a = constant acceleration, vt = velocity at time t, st = displacement at time t):

  1. \( \frac{343}{3} \)
  2. \( 0 \)
  3. \( \frac{24}{25} \)

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