Problem 106 - Calculus Early Transcendentals

Question

Cubic zero net area Consider the graph of the cubic \(y=x(x-a)(x-b),\) where \(0

Step-by-Step Solution

Answer

Based on the analysis and integration steps, we found that no possible relationship between \(a\) and \(b\) exists that would make the areas equal while both \(a\) and \(b\) are positive and \(a

Step 1: Analyze the cubic function

The given function is \(y = x(x-a)(x-b)\). Notice that when \(x=0\), \(y=0\). When \(0

Step 2: Write the area using integrals

To find the areas, we should integrate the given function over the specified intervals. Remember that the function is above the x-axis for \(0

Step 3: Set up the area equality equation and solve for \(a\) and \(b\)

The problem asks us to find the relationship between \(a\) and \(b\) such that the areas of these two regions are equal. So, we set up the equality: \(\int_0^a x(x-a)(x-b)\,dx = -\int_a^b x(x-a)(x-b)\,dx\) Since the integrands are the same, we can simplify this equation as: \(a (a-b) = -b (a-b)\) By simplification, we get: \(a (a-b) - b (a-b) = 0\) Factoring out \((a-b)\), we have: \((a-b)(a + b) = 0\) Since \(a\) and \(b\) are both positive and \(a < b\), we know that \((a-b)\) cannot be zero. Hence, the only solution is: \(a + b = 0\) However, this solution contradicts the initial condition of the problem that \(a\) and \(b\) are both positive. So there is no possible relationship between \(a\) and \(b\) that would make the areas equal under the problem's conditions.

Key Concepts

Definite Integrals

Understanding definite integrals is crucial when tackling problems involving areas under curves, especially with functions such as the cubic function. A definite integral, represented by the notation \(\int_a^b f(x)\,dx\), essentially computes the net area between the graph of \(f(x)\) and the \(x\)-axis over the interval from \(a\) to \(b\).

In the context of the cubic zero net area problem, definite integrals allow us to calculate the precise area of the regions bounded by the cubic function above and below the \(x\)-axis. When a function lies above the \(x\)-axis, the definite integral gives the area directly as a positive value. Conversely, when the function is below the \(x\)-axis, the integral yields a negative value. To find the positive area in such cases, we negate the integral, as depicted in the textbook exercise. This method ensures we can compare the magnitudes of areas regardless of their position relative to the \(x\)-axis.

Graph of Cubic Function

The graph of a cubic function, like \(y=x(x-a)(x-b)\), is characterized by its unique shape, which includes turns and possibly one or more inflection points. The specific features of the graph depend on the coefficients and the sign of the leading term. In our exercise, it's important to visualize the graph to understand how the function behaves above and below the \(x\)-axis between \(0\) and \(b\).

For \(0

Properties of Cubic Functions

Cubic functions bring a set of interesting properties that come into play when solving equations or analyzing graphs. These properties include the possibility of having one, two, or three real roots, and a shape that can intersect the \(x\)-axis up to three times.

Key Properties:

• End Behavior: Cubic functions will have opposite behavior at each end, heading to infinity in opposite directions as \(x\) increases or decreases without bound.
• Turning Points: They can have up to two turning points, which create the characteristic 's' shape of the function's graph.
• Inflection Point: There is one point where the curvature changes, known as the inflection point.

In the given exercise, these properties are crucial for understanding the regions where the function is positive or negative and thus affect the calculation of areas through definite integrals.

Solving Cubic Equations

Solving cubic equations can be challenging. They may require a variety of methods to find their roots, including factoring by grouping, synthetic division, or even graphing. However, the availability of a formulaic solution for cubic equations is less well-known compared to the quadratic formula.

The provided exercise showcases an approach that involves analyzing the behavior of the function's graph and setting up integral equations to determine when certain areas are equal. This approach, while not directly solving for roots, uses the concept of roots to understand where the function will cross the \(x\)-axis and hence inform our integral setup. It's an illustrative example of how understanding the behavior of cubic functions aids in solving the equations they present us with.

Previous

Problem 106

Next

Problem 107

Other exercises in this chapter

Problem 105

Substitution: scaling Another change of variables that can be interpreted geometrically is the scaling \(u=c x,\) where \(c\) is a real number. Prove and interp

View solution

Problem 106

If necessary, use two or more substitutions to find the following integrals. \(\int x \sin ^{4} x^{2} \cos x^{2} d x\left(\text {Hint}: \text { Begin with } u=x

View solution

Problem 107

If necessary, use two or more substitutions to find the following integrals. \(\int \frac{d x}{\sqrt{1+\sqrt{1+x}}}(\text {Hint: Begin with } u=\sqrt{1+x} .)\)

View solution

Problem 107

Maximum net area What value of \(b>-1\) maximizes the integral $$\int_{-1}^{b} x^{2}(3-x) d x ?$$

View solution

Key Concepts

Definite Integrals

Understanding definite integrals is crucial when tackling problems involving areas under curves, especially with functions such as the cubic function. A definite integral, represented by the notation \(\int_a^b f(x)\,dx\), essentially computes the net area between the graph of \(f(x)\) and the \(x\)-axis over the interval from \(a\) to \(b\).

In the context of the cubic zero net area problem, definite integrals allow us to calculate the precise area of the regions bounded by the cubic function above and below the \(x\)-axis. When a function lies above the \(x\)-axis, the definite integral gives the area directly as a positive value. Conversely, when the function is below the \(x\)-axis, the integral yields a negative value. To find the positive area in such cases, we negate the integral, as depicted in the textbook exercise. This method ensures we can compare the magnitudes of areas regardless of their position relative to the \(x\)-axis.

Graph of Cubic Function

The graph of a cubic function, like \(y=x(x-a)(x-b)\), is characterized by its unique shape, which includes turns and possibly one or more inflection points. The specific features of the graph depend on the coefficients and the sign of the leading term. In our exercise, it's important to visualize the graph to understand how the function behaves above and below the \(x\)-axis between \(0\) and \(b\).

For \(0

Properties of Cubic Functions

Cubic functions bring a set of interesting properties that come into play when solving equations or analyzing graphs. These properties include the possibility of having one, two, or three real roots, and a shape that can intersect the \(x\)-axis up to three times.

Key Properties:

• End Behavior: Cubic functions will have opposite behavior at each end, heading to infinity in opposite directions as \(x\) increases or decreases without bound.
• Turning Points: They can have up to two turning points, which create the characteristic 's' shape of the function's graph.
• Inflection Point: There is one point where the curvature changes, known as the inflection point.

In the given exercise, these properties are crucial for understanding the regions where the function is positive or negative and thus affect the calculation of areas through definite integrals.

Solving Cubic Equations

Solving cubic equations can be challenging. They may require a variety of methods to find their roots, including factoring by grouping, synthetic division, or even graphing. However, the availability of a formulaic solution for cubic equations is less well-known compared to the quadratic formula.

The provided exercise showcases an approach that involves analyzing the behavior of the function's graph and setting up integral equations to determine when certain areas are equal. This approach, while not directly solving for roots, uses the concept of roots to understand where the function will cross the \(x\)-axis and hence inform our integral setup. It's an illustrative example of how understanding the behavior of cubic functions aids in solving the equations they present us with.

Previous

Problem 106

Next

Problem 107

Other exercises in this chapter

Problem 105

Substitution: scaling Another change of variables that can be interpreted geometrically is the scaling \(u=c x,\) where \(c\) is a real number. Prove and interp

View solution

Problem 106

If necessary, use two or more substitutions to find the following integrals. \(\int x \sin ^{4} x^{2} \cos x^{2} d x\left(\text {Hint}: \text { Begin with } u=x

View solution

Problem 107

If necessary, use two or more substitutions to find the following integrals. \(\int \frac{d x}{\sqrt{1+\sqrt{1+x}}}(\text {Hint: Begin with } u=\sqrt{1+x} .)\)

View solution

Problem 107

Maximum net area What value of \(b>-1\) maximizes the integral $$\int_{-1}^{b} x^{2}(3-x) d x ?$$

View solution