Understanding the Area Between a Curve and the X-Axis: y x^2 - 4x from x -2 to x 0

Introduction

When dealing with curves and the X-axis, one of the common tasks is to calculate the area between the curve and the X-axis over a specified interval. In this article, we focus on the curve defined by the equation y x^2 - 4x and determine the area under this curve from x -2 to x 0.

Mathematical Analysis

Integral Calculation: To find the area between the curve y x^2 - 4x and the X-axis from x -2 to x 0, we need to evaluate the integral [A int_{-2}^{0} (x^2 - 4x) , dx.]

Step-by-Step Solution

Find the Antiderivative

The antiderivative of the function y x^2 - 4x can be found using the power rule of integration, which states that the antiderivative of x^n is (x^(n 1))/(n 1). Therefore, the antiderivative of x^2 is x^3/3, and the antiderivative of -4x is -2x^2. Combining these, we get [int (x^2 - 4x) , dx frac{x^3}{3} - 2x^2 C,]

Evaluate the Definite Integral

To evaluate the definite integral, we apply the limits of integration from -2 to 0 [A left[ frac{x^3}{3} - 2x^2 right]_{-2}^{0}.]

Substituting the upper limit 0 into the antiderivative, we get [frac{0^3}{3} - 2(0^2) 0,]

And substituting the lower limit -2 into the antiderivative, we get [frac{(-2)^3}{3} - 2(-2)^2 frac{-8}{3} - 2(4) frac{-8}{3} - 8 frac{-8}{3} - frac{24}{3} frac{-32}{3}.]

Since the area can never be negative, we take the absolute value of the result, which is [A left| frac{-32}{3} right| frac{32}{3}.]

Conclusion

The area between the curve y x^2 - 4x and the X-axis from x -2 to x 0 is (frac{32}{3}) square units), which is approximately (10.67) square units.

Key Takeaways

Understanding and applying the power rule of integration is crucial for calculating definite integrals. Interpreting the results in the context of area calculation ensures the result is always positive. Using antiderivatives effectively allows the solution of complex curve integration problems.

Additional Resources

For further practice and a deeper understanding of calculus, consider revisiting this tutorial on integration or exploring more problems in calculus textbooks.