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Introduction to Integrals. In this chapter we give an informal introduction to integration of functions of several variables. For functions of a single variable, the de nite Riemann integral is de ned for functions whose domain is a closed bounded interval.
The most generally useful and powerful integration technique re-mains Changing the Variable. The first Problems in this section provide additional practice changing variables to calculate integrals. As we develop more complicated and more specialized techniques for finding antiderivatives, your first thought should still be whether the integral
Use the basic integration formulas to find indefinite integrals. Use substitution to find indefinite integrals. Use substitution to evaluate definite integrals. Use integration to solve real-life problems. Review of Basic Integration Formulas.
Spring 2020. The point: Techniques for computing integrals are derived, using interpolation and piece-wise constructions (composite formulas). In addition, the asymptotic error series for the trapezoidal rule is introduced, enabling the use of Richardson extrapolation for integration.
The problem of integration is to find a limit of sums. The key is to work backward from a limit of differences (which is the derivative). We can integrate v.x/if it turns up as the derivative of another function f.x/. The integral of vDcos xis fDsin x. The integralof vDxis fD 1 2 x2. Basically, f.x/is an “antiderivative”.
an integral: we approximate the area under a curve by bounding it between rectangles. Later, we will learn more sophisticated methods of integration, but they are all based on this simple concept.
This chapter shows how to integrate functions of two or more variables. First, a double integral is defined as the limit of sums. Second, we find a fast way to compute it. The key idea is to replace a double integral by two ordinary "single" integrals. The double integral Sf f(x, y)dy dx starts with 1f(x, y)dy. For each fixed x we integ-
