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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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