Question

In Exercises 75-77, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the area of the region bounded by the graphs of \(f\) and \(g\) is \(1,\) then the area of the region bounded by the graphs of \(h(x)=f(x)+C\) and \(k(x)=g(x)+C\) is also 1.

Short answer

The statement is true that if the area of the region bounded by the graphs of \(f\) and \(g\) is \(1,\) then the area of the region bounded by the graphs of \(h(x)=f(x)+C\) and \(k(x)=g(x)+C\) is also 1. This is because the addition of a constant to a function translates the graph vertically, but does not change the shape of the graph or the area between it and another function.

Step-by-step solution

Understanding the Concept

The main concept of this problem relies on understanding how functions change when a constant is added or subtracted. When a constant \(C\) is added to a function, the graph of the function shifts up if \(C>0\) or shifts down if \(C<0\), but the shape of the graph remains the same. Hence, the area between two functions should remain the same after adding the same constant to both because the vertical distance between the functions at any point \(x\) stays the same.

Working on the Statement

Let's illustrate this with an example: Consider two functions \(f(x) = x^2\) and \(g(x) = x\). We can calculate the area between them, \( \int (f(x) - g(x)) dx \), to be \( \frac{1}{3} \). Now, if we add 1 to both functions we have \(h(x) = x^2 +1\) and \(k(x) = x +1\). The area between them, calculated the same way \( \int (h(x) - k(x)) dx \), is still \( \frac{1}{3} \). This supports that the statement is true.

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the accumulation of quantities and the areas under and between curves. It deals with integrals and their properties. For example, when we wish to find the area between two curves described by functions, we use integral calculus to evaluate it. This is done by taking the integral of the absolute difference between the given functions over the interval of interest.

When students approach problems involving the area between curves, they usually calculate the definite integral of this difference. The value of the definite integral represents the total 'accumulated' area between the graphs over the specified range. If we consider the question from the textbook, which involves finding the area between two functions that are altered by adding a constant, integral calculus provides a systematic way to prove whether the area remains the same or changes.

Let's look at an illustration: To find the area between the curves of functions \( f(x) \) and \( g(x) \), we evaluate the definite integral \( \int_a^b |f(x) - g(x)| dx \), where \(a\) and \(b\) define the interval on the \(x\)-axis over which we are interested. The integral accumulates the vertical differences between \( f(x) \) and \( g(x) \) at every point, summing up to the total area.

Functions and Graphs

The relationship between functions and their corresponding graphs is foundational in understanding the concepts in calculus, especially when determining areas between curves. A function can be visually represented as a curve on a graph, showing the different outputs \( f(x) \) for inputs \( x \).

The graph can provide insight into the behavior of the function, such as where it increases, decreases, where it reaches a maximum or minimum value, and how it interacts with other functions graphed on the same coordinate plane. When examining the area between two functions, their graphs can make it easier to see which function is above or below over a certain interval. The points where the graphs intersect are often critical, as they define the limits of integration when calculating the area between them.

Utilizing graphs also aids in understanding the symmetrical aspects of functions, which can simplify calculations. For instance, if two functions are symmetrical across the \(y\)-axis, their areas on either side of the axis might cancel each other out, resulting in a net area of zero. The textbook exercise deals with understanding the impact of shifting graphs vertically on the area between them, a concept that is easier to comprehend when visualized on graphs.

Shifting Graphs Vertically

Shifting graphs vertically is a transformation that occurs when a constant is added to or subtracted from every output value of a function. This type of transformation does not affect the 'shape' of the graph but simply moves it up or down along the \( y \)-axis.

When working with the area between curves, if we apply the same vertical shift to both functions, the vertical distance between their graphs at any point along the \( x \)-axis remains constant. This means that the area enclosed by the two curves remains unchanged, despite the shift. Consider functions \( f(x) \) and \( g(x) \) with a constant area between them. If we define new functions \( h(x) = f(x) + C \) and \( k(x) = g(x) + C \), where \( C \) is a constant, the vertical shift happens to both functions equally.

In the example given in the textbook solution, adding the same constant to both \( f(x) \) and \( g(x) \) does not change the area between the curves. This is a key observation students should make, as it showcases a property of linearity in integrating the difference between two functions. The understanding of vertical shifts also enhances the comprehension of various other transformations, such as reflections and horizontal shifts, which likewise affect graphing functions.