82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.
b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.

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Identify the function to integrate: the area under the curve y = x e^{-a x} from x = 0 to x = 4 is given by the definite integral \( \int_0^4 x e^{-a x} \, dx \), where \( a > 0 \).

Set up the integral explicitly: \( \text{Area} = \int_0^4 x e^{-a x} \, dx \). This integral represents the area bounded by the curve and the x-axis over the interval [0,4].

Use integration by parts to solve the integral. Let \( u = x \) and \( dv = e^{-a x} dx \). Then, compute \( du = dx \) and \( v = \int e^{-a x} dx = -\frac{1}{a} e^{-a x} \).

Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions to get \( \int_0^4 x e^{-a x} dx = \left. -\frac{x}{a} e^{-a x} \right|_0^4 + \frac{1}{a} \int_0^4 e^{-a x} dx \).

Evaluate the remaining integral \( \int_0^4 e^{-a x} dx \), which is \( \left. -\frac{1}{a} e^{-a x} \right|_0^4 \). Then, combine all terms to express the area in terms of \( a \) without calculating the numerical value.

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주요 개념

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

Definite Integral and Area Under a Curve

The definite integral of a function over an interval represents the net area between the curve and the x-axis. For positive functions, this corresponds to the actual area bounded by the curve, the x-axis, and the vertical lines at the interval endpoints. Calculating this integral gives the total area under the curve on the specified interval.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and is especially useful when integrating expressions like x * e^(-ax). The formula is ∫u dv = uv - ∫v du, where u and dv are parts of the original integrand chosen to simplify the integral.

Exponential Decay and Parameter Effects

The function y = x * e^(-ax) combines a linear term and an exponential decay term. The parameter 'a' controls the rate of decay: larger 'a' values cause the function to decrease more rapidly. Understanding how 'a' affects the shape and area under the curve is crucial for evaluating the integral and interpreting the graph.

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.