Areas of regions Find the area of the following regions.                                                                                                                   
                                                                                                                                                                 The region bounded by the graph of ƒ(𝓍) = (𝓍―4)⁴ and the 𝓍-axis between and 𝓍 = 2 and 𝓍= 6

Verified step by step guidance

1. Identify the function and the interval: The function is given as \(f(x) = (x - 4)^4\), and the region is bounded between \(x = 2\) and \(x = 6\) along the \(x\)-axis.
2. Determine the nature of the function on the interval: Since \((x - 4)^4\) is always non-negative for all real \(x\), the graph lies above or on the \(x\)-axis between \(x=2\) and \(x=6\).
3. Set up the definite integral to find the area: The area under the curve and above the \(x\)-axis from \(x=2\) to \(x=6\) is given by the integral \(\int_{2}^{6} (x - 4)^4 \, dx\).
4. Apply the power rule for integration: Recall that \(\int (x - a)^n \, dx = \frac{(x - a)^{n+1}}{n+1} + C\). Use this to write the antiderivative of \((x - 4)^4\).
5. Evaluate the definite integral: Substitute the upper limit \(x=6\) and the lower limit \(x=2\) into the antiderivative expression and find the difference to get the area.

Key Concepts

Definite Integral as Area Under a Curve

The definite integral of a function between two points gives the net area between the graph and the x-axis over that interval. For functions above the x-axis, this corresponds to the actual area. Calculating this integral helps find the exact area of the region bounded by the curve and the axis.

Properties of Even-Powered Polynomial Functions

Functions like ƒ(x) = (x - 4)^4 are always non-negative because even powers eliminate negative values. This ensures the graph lies on or above the x-axis, simplifying area calculations since the integral directly represents the area without needing to consider negative parts.

Setting Integration Limits Based on the Region

The limits of integration correspond to the x-values that bound the region of interest. Here, the area is between x = 2 and x = 6, so these values are used as the lower and upper limits in the definite integral to calculate the area under the curve within this interval.