Final answer:
Definite integrals are computed by determining the antiderivative of a function, applying the limits of integration, and taking the difference between the upper and lower bounds. For complex multivariable functions, such as surface and line integrals, parametrizing the subject of integration and then reducing it to single-variable integrals is necessary.
Step-by-step explanation:
Definite integrals are a fundamental tool in calculus used to calculate the accumulation of quantities, such as areas, volumes, and total values, over an interval. The process of finding the definite integral involves several steps:
- Determining the function to be integrated.
- Identifying the limits of integration, which are the bounds of the interval over which the accumulation is to be calculated.
- Evaluating the antiderivative of the function with respect to the variable of integration.
- Subtracting the value of the antiderivative at the lower limit from the value at the upper limit, a process known as the Fundamental Theorem of Calculus.
An example is integrating a function f(x) from x1 to x2, which graphically represents the area under the curve of f(x) between those two points. This corresponds to the sum of infinitesimally small areas, f(x) dx, between x1 and x2 as depicted in Figure 7.8.
To calculate surface integrals and line integrals, one must parametrize the surface or curve and then reduce the multiple integrals to single-variable integrals. For vector integrals, such as those appearing in physics, aligning contributions with the correct orientation can be challenging. Evaluation strategies often involve choosing the right coordinate system and expressing the integrand in terms of these coordinates.