14–25. {Use of Tech} Areas of regions Determine the area of the given region.


The region bounded by y = x²,y = 2x²−4x, and y = 0

Verified step by step guidance

1. First, identify the curves that bound the region: \( y = x^2 \), \( y = 2x^2 - 4x \), and \( y = 0 \). We want to find the area enclosed by these curves.
2. Next, find the points of intersection between the curves to determine the limits of integration. Start by setting \( y = x^2 \) equal to \( y = 2x^2 - 4x \) and solve for \( x \): \[ x^2 = 2x^2 - 4x \]
3. Rearrange the equation to isolate terms: \[ 0 = 2x^2 - 4x - x^2 = x^2 - 4x \] Then factor: \[ x(x - 4) = 0 \] So the intersection points are at \( x = 0 \) and \( x = 4 \).
4. Determine which curve is on top (greater \( y \) value) between \( x = 0 \) and \( x = 4 \) by testing a point in the interval, for example \( x = 1 \). This will help set up the integral for the area between the curves.
5. Set up the integral for the area between the curves and above \( y = 0 \) as: \[ \text{Area} = \int_{0}^{4} \left( \text{top function} - \text{bottom function} \right) \, dx \] where the top and bottom functions are determined from the previous step.

Key Concepts

Finding Points of Intersection

To determine the area bounded by curves, first find where the curves intersect by setting their equations equal. These intersection points define the limits of integration and help identify the region enclosed by the curves.

Setting Up Definite Integrals for Area

The area between curves is found by integrating the difference of the functions over the interval defined by their intersection points. Specifically, integrate the upper function minus the lower function with respect to x to calculate the enclosed area.

Understanding the Role of the x-axis (y=0)

Since the region is bounded by y=0 (the x-axis), it acts as a boundary line. Recognizing when the x-axis forms part of the boundary helps in correctly setting up the integral and determining which parts of the curves contribute to the enclosed area.