Area Under the Curve Question - None

Question

If the area bounded by the graph of y = xe^−ax (a > 0) and the abscissa axis is 1/9, then the value of a is equal to

Accepted Answer

Solution: The curve y = xe^−ax crosses the x-axis at x = 0 and approaches 0 as x → ∞.

The area bounded by the curve and the x-axis is:

$$A = \int_{0}^{\infty} x e^{-ax} dx$$

Using integration by parts: $$u = x, \, dv = e^{-ax}dx$$

$$A = \left[-\frac{x e^{-ax}}{a}\right]_{0}^{\infty} + \int_{0}^{\infty} \frac{e^{-ax}}{a} dx = 0 + \frac{1}{a^2}$$

Given that $A = \frac{1}{9}$, we have:

$$\frac{1}{a^2} = \frac{1}{9} \Rightarrow a^2 = 9 \Rightarrow a = 3$$

∴ Answer is T (3).

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