Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.
ƒ(𝓍) = 𝓍⁴ ― 𝓍² on [―1, 1]

Verified step by step guidance

1. Step 1: Understand the problem. The net area is the integral of the function ƒ(𝓍) = 𝓍⁴ ― 𝓍² over the interval [―1, 1]. The area, on the other hand, is the integral of the absolute value of ƒ(𝓍) over the same interval. Sketching the graph of ƒ(𝓍) will help identify where the function is positive and negative.
2. Step 2: Analyze the function ƒ(𝓍) = 𝓍⁴ ― 𝓍². Factorize the function to find its roots: ƒ(𝓍) = 𝓍²(𝓍² ― 1) = 𝓍²(𝓍 ― 1)(𝓍 + 1). The roots are 𝓍 = ―1, 𝓍 = 0, and 𝓍 = 1. These are the points where the function intersects the 𝓍-axis.
3. Step 3: Determine where the function is positive and negative. By testing intervals between the roots (e.g., [―1, 0], [0, 1]), you can determine the sign of ƒ(𝓍). This will help in calculating the net area and the absolute area.
4. Step 4: Compute the net area. Set up the integral ∫[―1, 1] ƒ(𝓍) d𝓍 = ∫[―1, 1] (𝓍⁴ ― 𝓍²) d𝓍. Evaluate this integral directly, keeping in mind that the negative contributions will reduce the total value.
5. Step 5: Compute the area. Set up the integral for the absolute value of ƒ(𝓍): ∫[―1, 1] |ƒ(𝓍)| d𝓍. Split the integral into regions where ƒ(𝓍) is positive and negative, and evaluate each separately. For example, ∫[―1, 0] |ƒ(𝓍)| d𝓍 and ∫[0, 1] |ƒ(𝓍)| d𝓍. Add these results to find the total area.

Key Concepts

Net Area

Net area refers to the total area between a curve and the x-axis, taking into account the sign of the function. When the function is above the x-axis, the area is positive, and when it is below, the area is negative. To find the net area over a specific interval, one must calculate the definite integral of the function over that interval, which effectively sums these areas.

Definite Integral

The definite integral of a function over an interval provides a way to calculate the area under the curve of that function between two points. It is denoted as ∫[a, b] f(x) dx, where 'a' and 'b' are the limits of integration. The Fundamental Theorem of Calculus connects differentiation and integration, allowing us to evaluate the definite integral using antiderivatives.

Area Calculation

To calculate the area of a region bounded by a curve and the x-axis, one must consider the absolute value of the function's output. This means that regardless of whether the function is above or below the x-axis, the area is always treated as positive. The area can be found by integrating the absolute value of the function over the specified interval, ensuring that all contributions to the area are counted positively.