Problem 83 - Calculus of a Single Variable

Question

True or False? In Exercises \(83-86\) , 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 \(1 .\)

Step-by-Step Solution

Answer

The statement is true. The constant \(C\) like in functions \(h(x)\) and \(k(x)\) does not influence the area between the two curves.

Step 1: Analyzing the statement

The relationships between the functions \(f\), \(g\) and \(h\), \(k\) is that each of them is a vertical shift by the constant \(C\) of the corresponding original functions. Hence, it is necessary to understand if vertica shift has any effect on the bounded area between two functions.

Step 2: Applying integral properties

The definite integral of \(f\) over an interval represents the signed area of the region between the function and the x-axis. If the function \(f(x)\) is translated vertically by a constant \(C\), the resulting area is not changed. The same applies to the function \(g(x)\). Therefore, the area between the function \(f\) and \(g\) will be equal to the area between \(h(x) = f(x) + C\) and \(k(x) = g(x) + C\) because the vertical shift doesn't change the definite integral.

Step 3: Verification of the statement

The area between the function \(f\) and \(g\) is calculated as the definite integral of \(g(x) - f(x)\) over a certain interval. If we now examine the area between \(h\) and \(k\), we would calculate it as the definite integral of \(k(x) - h(x)\) over the same interval, but as both \(h(x)\) and \(k(x)\) are the vertical translations of \(f(x)\) and \(g(x)\) respectively, the subtraction would eliminate the constant \(C\), rendering the statement true.

Key Concepts

Definite Integral

The concept of the definite integral is crucial in understanding how to compute areas under curves. When you have a function, say \(f(x)\), the definite integral of this function over an interval \([a, b]\) gives you the net signed area between the curve and the x-axis. This is captured in the notation:
\[\int_{a}^{b} f(x) \, dx\]

• The "definite" part means you are dealing with specific limits, here from \(a\) to \(b\).
• This integral evaluates the total accumulation of area, considering that area above the x-axis is positive and below is negative.

This understanding helps when comparing areas between two functions, where the area is expressed as:
\[\int_{a}^{b} [g(x) - f(x)] \, dx\]This computes the area between the curves of \(g\) and \(f\) over the same interval.

Vertical Shift

The vertical shift of functions is a transformation where a constant, \(C\), is added to the entire function. Consider the function \(f(x)\). If we add a constant \(C\), it transforms into \(h(x) = f(x) + C\).

• This shift moves every point on the graph of the function up or down by \(C\) units.
• Despite this shift, the shape and orientation of the graph do not change.

Applying this to the bounded area, if \(f(x)\) and \(g(x)\) define a region, a vertical shift to form \(h(x) = f(x) + C\) and \(k(x) = g(x) + C\) does not alter the region's bounded area. This is because both functions move similarly, maintaining their original relative distance and orientation.

Bounded Area

A bounded area is the region between two graph functions over a specified interval.
When calculating this area, the definite integral is used to find the space between functions, say \(f(x)\) and \(g(x)\), as:
\[\int_{a}^{b} [g(x) - f(x)] \, dx\]

• This integral gives the area between \(f\) and \(g\), where \(g(x)\) is the upper function and \(f(x)\) is the lower.
• A true understanding of bounded areas ensures that any transformations or adjustments to functions are considered wisely.

In our context, even if functions \(f(x)\) and \(g(x)\) are shifted vertically, the bounded area remains unaffected because the shift is uniform.

Graph of Functions

Graphs of functions provide a visual representation of equations, illustrating how the outputs (y-values) change with the inputs (x-values).

• This helps in visually understanding concepts such as area under the curve, especially in calculus.
• When sketching functions like \(f(x)\) or \(g(x)\), any transformations such as vertical shifts are clearly visible.

In examining changes such as vertical shifts, the graph demonstrates that while each point on the graph moves up or down, the space between two parallel shifts—representing functions like \(h(x)\) and \(k(x)\)—remains constant.
Understanding these visuals aids in grasping the unchanged nature of the bounded area between shifted functions.