Problem 24 - Calculus Single Variable

Question

Find the area of the region that is bounded by the graphs of \(y=f(x)\) and \(y=g(x)\) for \(x\) between the abscissas of the two points of intersection. $$ f(x)=x^{2} \quad g(x)=-2 x^{2}-15 x-18 $$

Step-by-Step Solution

Answer

The area of the region bounded by the curves is 0.5.

Step 1: Find the Points of Intersection

To find the points where the graphs of \(y = f(x)\) and \(y = g(x)\) intersect, set \(f(x) = g(x)\). This gives \(x^2 = -2x^2 - 15x - 18\). Moving all terms to one side, we have \(3x^2 + 15x + 18 = 0\). Let's solve this quadratic equation using the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), where \(a = 3\), \(b = 15\), \(c = 18\).

Step 2: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Here, \(b = 15\), \(a = 3\), and \(c = 18\). Thus, the discriminant is \(15^2 - 4(3)(18) = 225 - 216 = 9\). Since the discriminant is positive, there are two real roots.

Step 3: Solve for the Intersection Points

Apply the quadratic formula: \(x = \frac{{-15 \pm \sqrt{9}}}{6}\). Simplifying this, we find the roots: \(x = \frac{{-15 + 3}}{6} = -2\) and \(x = \frac{{-15 - 3}}{6} = -3\). So, the points of intersection are at \(x = -2\) and \(x = -3\).

Step 4: Set Up the Integral for Area

The area between the curves from \(x = -3\) to \(x = -2\) is given by the integral: \[ A = \int_{-3}^{-2} (g(x) - f(x)) \, dx \]. Substituting the functions, we have: \( A = \int_{-3}^{-2} (-2x^2 - 15x - 18 - x^2) \, dx \). Simplify the expression inside the integral: \(-3x^2 - 15x - 18\).

Step 5: Integrate the Function

Integrate the simplified function: \(\int (-3x^2 - 15x - 18) \, dx\). The indefinite integral is \(-x^3 - \frac{15}{2}x^2 - 18x + C\), where \(C\) is a constant of integration.

Step 6: Evaluate the Definite Integral

Compute \( A = \left. \left(-x^3 - \frac{15}{2}x^2 - 18x \right) \right|_{-3}^{-2}\). Evaluating, we get: For \(x = -2\), the value is \(-(-2)^3 - \frac{15}{2}(-2)^2 - 18(-2) = -(-8) - 30 - 36 = 8 - 30 + 36 = 14\).For \(x = -3\), the value is \(-(-3)^3 - \frac{15}{2}(-3)^2 - 18(-3) = -(-27) - \frac{135}{2} + 54 = 27 - 67.5 + 54 = 13.5\).The area is \(14 - 13.5 = 0.5\).

Step 7: Conclusion

The area of the region bounded by the curves \(y = f(x)\) and \(y = g(x)\) from \(x = -3\) to \(x = -2\) is \(0.5\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the roots of a quadratic equation in the form \( ax^2 + bx + c = 0 \). In this exercise, we utilized the quadratic formula to determine the points where two curves intersect. The formula is given by:\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]Using the coefficients from the combined equation \( 3x^2 + 15x + 18 = 0 \), we identified:

• \(a = 3\)
• \(b = 15\)
• \(c = 18\)

By substituting these values into the quadratic formula, we can find the specific values of \(x\) where two graphs intersect. This process helps to pinpoint the boundaries needed for other calculations, such as integration.

Definite Integral

A definite integral represents the total area under a curve between two specific limits. For this problem, the area is bounded between the intersection points of two curves. The general formula for a definite integral is:\[ \int_{a}^{b} f(x) \, dx \]In our exercise, we calculated:\[ A = \int_{-3}^{-2} (-3x^2 - 15x - 18) \, dx \]This integral calculates the area of the region described by the difference of the curves \(y = g(x)\) and \(y = f(x)\) over the interval from \(x = -3\) to \(x = -2\). Integrals play a crucial role in determining the exact area of complex shapes bounded by curves.

Points of Intersection

Finding the points of intersection involves determining where two functions share the same value. For curves, this happens when their equations equalize each other. In our case:\[ x^2 = -2x^2 - 15x - 18 \]Solving this quadratic equation revealed the intersection points at \(x = -2\) and \(x = -3\). These points demarcate the interval where the area between the curves is calculated. By identifying these points accurately, we ensure that our calculations for area are precise and that we correctly interpret the geometrical picture entailed by the functions.

Discriminant

The discriminant is an important part of the quadratic formula, helping to determine the nature of the roots without solving the entire equation. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is\[ b^2 - 4ac \]In this exercise, the discriminant is calculated as:\[ 15^2 - 4 \times 3 \times 18 = 225 - 216 = 9 \]A positive discriminant like 9 indicates that there are two real and distinct solutions. This aligns with our solution, where the intersection points were different \(x\) values. The discriminant is a quick check to forecast how many points of intersection, or roots, an equation might have before solving it out completely.