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

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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.

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 $x=6$.

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 $\int_{2}^{6} (x - 4)^4 \, dx$.

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 $(x - 4)^4$.

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.

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Key Concepts

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

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.

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