Problem 8 - Calculus Early Transcendentals

Question

Why can the constant of integration be omitted from the antiderivative when evaluating a definite integral?

Step-by-Step Solution

Answer

Answer: The constant of integration can be omitted when evaluating a definite integral because it cancels out during the process of subtracting the antiderivative at the limits of integration. This general result applies to all definite integrals, making it unnecessary to include the constant of integration when evaluating them.

Step 1: Understand Antiderivatives and Constants of Integration

An antiderivative of a function is another function that represents the integral of the original function. The constant of integration is introduced because there can be an infinite number of antiderivatives for a single function. This constant represents the possible vertical shifts of the antiderivative. For example, the antiderivative of f(x) = 2x is F(x) = x^2 + C, where C is the constant of integration.

Step 2: Understand Definite Integrals

A definite integral, written as ∫[a, b]f(x)dx, represents the signed area between the curve defined by the function f(x) and the x-axis, over the interval [a, b].

Step 3: Evaluate a Definite Integral using Antiderivative

To evaluate a definite integral, we can follow the Fundamental Theorem of Calculus. It tells us to first find the antiderivative of the function, and then subtract the antiderivative evaluated at the lower limit of integration (a) from the antiderivative evaluated at the upper limit of integration (b). Mathematically: ∫[a, b]f(x)dx = F(b) - F(a)

Step 4: Observe the Role of the Constant of Integration

Now, let's analyze the role of the constant of integration within the definite integral evaluation. Using the antiderivative found earlier, F(x) = x^2 + C: ∫[a, b]2xdx = (b^2 + C) - (a^2 + C) Notice that the constant of integration, C, appears in both terms, and when we subtract them, the C terms cancel each other out: (b^2 + C) - (a^2 + C) = b^2 - a^2

Step 5: Conclusion

The constant of integration can be omitted from the antiderivative when evaluating a definite integral because it cancels out during the process of subtracting the antiderivative at the limits of integration. This is a general result that applies to all definite integrals, which is why we don't need to include the constant of integration when evaluating them.