Cubic zero net area Consider the graph of the cubic y = 𝓍 (𝓍― a) (𝓍― b), where 0 < a < b. Verify that the graph bounds a region above the 𝓍-axis, for 0 < 𝓍 < a , and bounds a region below the 𝓍-axis, for a < 𝓍 < b. What is the relationship between a and b if the areas of these two regions are equal? 

Verified step by step guidance

1. First, identify the roots of the cubic function \(y = x(x - a)(x - b)\), which are at \(x = 0\), \(x = a\), and \(x = b\), with \$0 < a < b$.
2. Next, analyze the sign of the function on the intervals determined by the roots: for \$0 < x < a\(, check the sign of each factor to confirm that \)y > 0\(, and for \)a < x < b\(, verify that \)y < 0$.
3. Set up the definite integrals representing the areas bounded by the curve and the \(x\)-axis: the area above the \(x\)-axis is \(\int_0^a x(x - a)(x - b) \, dx\), and the area below the \(x\)-axis is \(-\int_a^b x(x - a)(x - b) \, dx\) (note the negative sign to make the area positive).
4. Expand the cubic polynomial \(x(x - a)(x - b)\) to a standard polynomial form to simplify integration: \(x(x - a)(x - b) = x^3 - (a + b)x^2 + abx\).
5. Integrate both expressions and set the two areas equal to each other, then solve the resulting equation for the relationship between \(a\) and \(b\).

Key Concepts

Sign of a Cubic Function on Intervals

A cubic function with roots at 0, a, and b changes sign at each root. By analyzing the factors (x), (x - a), and (x - b) over intervals defined by these roots, we determine where the function is positive or negative. This helps identify regions above or below the x-axis, crucial for setting up area calculations.

Definite Integral as Net Area

The definite integral of a function over an interval gives the net area between the graph and the x-axis, counting areas below the axis as negative. To find the total area bounded above or below the axis, we integrate over the respective intervals and consider absolute values when comparing areas.

Equating Areas to Find Parameter Relationships

When two regions bounded by a curve have equal areas, setting their definite integrals equal (considering sign) allows us to find relationships between parameters like a and b. Solving these integral equations reveals conditions under which the areas above and below the x-axis are equal.