Areas of regions Find the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval.

ƒ(𝓍) = 𝓍³ ― 1 on [―1, 2]

Verified step by step guidance

Step 1: Understand the problem. You are tasked with finding the area of the region bounded by the graph of ƒ(𝓍) = 𝓍³ - 1 and the 𝓍-axis over the interval [―1, 2]. This involves integrating the function over the given interval and accounting for areas above and below the 𝓍-axis.

Step 2: Identify where the function crosses the 𝓍-axis. Set ƒ(𝓍) = 𝓍³ - 1 equal to 0 and solve for 𝓍. This gives the points where the function changes sign, which are important for splitting the integral into subintervals. Solve:

x^{3} - 1 = 0

. The solution is

x = 1

Step 3: Split the integral into subintervals based on the root found in Step 2. The function changes sign at 𝓍 = 1, so divide the interval [―1, 2] into [―1, 1] and [1, 2]. On [―1, 1], the function is below the 𝓍-axis, and on [1, 2], the function is above the 𝓍-axis.

Step 4: Set up the integrals for each subinterval. For the area below the 𝓍-axis on [―1, 1], take the absolute value of the integral:

- \int_{- 1} 1 x^{3} - 1 d x

. For the area above the 𝓍-axis on [1, 2], compute:

\int_{1} 2 x^{3} - 1 d x

Step 5: Combine the results. Add the absolute value of the integral on [―1, 1] to the integral on [1, 2] to find the total area. This ensures that areas below the 𝓍-axis are treated as positive contributions to the total area.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

The definite integral of a function over a specific interval represents the net area between the graph of the function and the x-axis. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. In this context, the definite integral will help determine the area bounded by the curve ƒ(x) = x³ - 1 and the x-axis from x = -1 to x = 2.

Area Under the Curve

The area under the curve refers to the total area between the graph of a function and the x-axis over a given interval. If the function is above the x-axis, the area is positive, while if it is below, the area is considered negative. To find the total area, one must account for both positive and negative areas, which may involve taking the absolute value of areas where the function is below the x-axis.

Critical Points and Sign Analysis

Critical points are values of x where the derivative of the function is zero or undefined, indicating potential local maxima, minima, or points of inflection. Analyzing the sign of the function at these points helps determine where the function is above or below the x-axis. For the function ƒ(x) = x³ - 1, finding critical points will assist in understanding the behavior of the graph and identifying the intervals contributing to the area calculation.

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