Exploring the Area Under the Curve y x^3 - 4x^2 - 3x: A Detailed Guide
This article delves into the calculations of the area under the curve given by the polynomial function yx3-4x2-3x from x0 to x3. We will cover the process of finding the x-intercepts, determining the extrema, and integrating the function to find the area.
1. Finding the x-intercepts
To find the x-intercepts, we set y0 in the given equation, leading to the equation x3-4x2-3x0. Factoring, we get:
[mathbf{x(x-1)(x 3) 0}]Thus, the x-intercepts are:
x0 x1 x-3For the given interval, the x-intercepts that fall within are x0 and x1.
2. Determining the Extrema
We find the extrema by taking the first derivative of the function:
[mathbf{y'} 3x^2 - 8x}]Setting y'0, we find the x-values of the extrema:
[mathbf{3x^2 - 8x 0}] [mathbf{x(3x - 8) 0}]Thus, the x-values are:
x0 x83Substituting these x-values into the original equation gives us the y-values:
y0 for x0 y-5927127 for x833. Calculating the Area Between the Curve and the x-axis
To find the area under the curve between x0 and x3, we integrate the function:
[mathbf{A int_{0}^{1} (x^3 - 4x^2 - 3x) , dx - int_{1}^{3} (x^3 - 4x^2 - 3x) , dx}]Evaluating these integrals, we get:
[mathbf{A left[frac{x^4}{4} - frac{4x^3}{3} - frac{3x^2}{2}right]_{0}^{1} - left[frac{x^4}{4} - frac{4x^3}{3} - frac{3x^2}{2}right]_{1}^{3}}]Simplifying these expressions, we find:
[mathbf{A left(frac{1}{4} - frac{4}{3} - frac{3}{2}right) - left(frac{81}{4} - frac{108}{3} - frac{27}{2}right) left(frac{1}{4} - frac{4}{3} - frac{3}{2}right)}]Further simplification gives:
[mathbf{A frac{1}{4} - frac{4}{3} - frac{3}{2} - frac{81}{4} frac{108}{3} frac{27}{2} frac{1}{4} - frac{4}{3} - frac{3}{2}}]This simplifies to:
[mathbf{A -frac{431}{12} 36} mathbf{frac{25}{12}}.Unit:square]The total area under the curve between x0 and x3 is approximately 2.08 square units.
Conclusion
The problem of finding the area under the curve y x^3 - 4x^2 - 3x from x 0 to x 3 requires the use of calculus to identify x-intercepts, extrema, and then integrate the function. This method ensures accuracy and a deeper understanding of polynomial functions and their behavior.