Question

Writing to Learn Is the following statement true, sometime true, or never true? The area of the region between the graphs of the continuous functions $$y=f(x)$$ and $$y=g(x)$$ and the vertical lines \(x=a\) and $x=b(x

Short answer

The original statement is 'sometimes' true - specifically, when function \(f(x)\) is consistently above \(g(x)\) on the interval [\(a, b\)]. For a more universally true statement, it should be revised to: 'The area between the regions of the continuous function graphs \(y = f(x)\) and \(y = g(x)\) and the vertical lines \(x = a\) and \(x = b\) (where \(x<b\)) is equal to \(\int_{a}^{b}|f(x) - g(x)| dx\)'.

Step-by-step solution

Understanding the statement

The statement being looked at asserts the area between the continuous functions \(y=f(x)\) and \(y=g(x)\) between \(x=a\) and \(x=b\), where \(x<b\), is equal to \(\int_{a}^{b}[f(x)-g(x)] dx\). This implies the area under the graph of \(f(x)\) subtracted by the area under the graph of \(g(x)\) from \(x=a\) to \(x=b\).

Evaluating the statement for continuous functions

The fundamental theory of calculus states that for a continuous real-valued function on a closed interval, the integral of a function can be interpreted as an area under the curve. So, for a continuous function, the \(\int_{a}^{b}[f(x)-g(x)] dx\) calculates the net area between the two functions. That means it can give a positive or negative result, depending on which function lies above the other.

Refining the statement

From the evaluation, it is clear that IF \(f(x)\) is always above \(g(x)\) in the interval \([a, b]\), then the statement is always true. But if this isn't the case, we could have a negative value for the result. This is because, in essence, this integral measures the net area, not the total area. To calculate the total area, irrespective of which graph is 'above' or 'below', we should use the absolute difference.

Final Verification and Revision

Upon understanding the conditions under which the statement would hold true, the original statement is not always correct. It would be more accurate to say: \textit{The area of the region between the graphs of the continuous functions \(y=f(x)\) and \(y=g(x)\) and the vertical lines \(x=a\) and \(x=b\), where \(x<b\), is equal to \(\int_{a}^{b}|f(x) - g(x)| dx\)}, because this integral denotes the positive difference between the two functions, resulting in the total area between the two curves.

Key concepts

Area Between Curves

When we talk about finding the area between two curves, we're usually referring to calculating the region that is enclosed by the graphs of two functions on a given interval. Imagine shading the space on a graph between two lines from one point to another; that area is what we aim to determine with definite integration. In the context of our exercise, the area between the curves represented by the functions y=f(x) and y=g(x) is specifically the region that lies between these two curves from x=a to x=b.

Now, it's crucial to realize that we subtract the values of g(x) from f(x) before integrating because we're effectively 'stacking' one function on top of the other and looking at the space between them. This approach only reflects the actual area when f(x) is always above g(x). If their positions interchange, or if they intersect, the formula needs to be adjusted to account for the absolute value, ensuring we measure the total area enclosed between them, not just the net space.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a pivotal concept that bridges the gap between calculus's two major branches: differentiation and integration. The theorem is split into two parts: the first part provides an antiderivative for a continuous function, while the second part states that the definite integral of a continuous function over [a, b] is equal to the difference between its antiderivative evaluated at b and a.

In simpler terms, the theorem tells us that integration can be used to calculate the accumulated value, such as area, of a function's rate of change (its derivative) over a certain interval. It underlines that if you know how to differentiate a function, you can also determine the area under its curve from point A to point B.

Continuous Functions

A function is said to be continuous at a point if you can draw its graph through that point without lifting your pen. For the more mathematically savvy, a function f(x) is continuous at point x=c if lim_(x->c) f(x) = f(c). In the context of our exercise, the functions f(x) and g(x) are both assumed to be continuous over the interval from x=a to x=b.

The significance of continuity in the context of integration can't be overstated. Continuous functions are 'nice' in the sense that they don't have gaps or jumps; they're smooth and well-behaved, which is essential for the Fundamental Theorem of Calculus to hold true. When we're assured that our functions don't have any discontinuities in the interval of interest, we can confidently apply the rules of integration to find area, average values, and other important concepts within calculus.

Integration

Understanding Integration

At its heart, integration is a method of adding up pieces to find the whole, commonly used to find areas under curves, among other things. When you integrate a function, you're essentially summing an infinite number of infinitesimally thin rectangles under the curve of the function. These rectangles may represent tiny slices of area, and when added together, provide the total area under the curve between two points.

Application in Calculating Areas

Regarding our exercise, integration allows us to calculate not just the area under a single curve, but also the area between two curves. Which, as we've discussed, can be found using the formula \(\int_{a}^{b}[f(x)-g(x)] dx\) with the caveat of incorporating the absolute value in cases where one function dips below the other within our interval of interest.