Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].

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Identify the function and the interval for which you need to find the area. Here, the function is \(g(x) = \frac{x^{2}}{\sqrt{9 - x^{2}}}\) and the interval is \([0, 2]\).

Recall that the area under the curve of a function \(g(x)\) from \(a\) to \(b\) is given by the definite integral \(\int_{a}^{b} g(x) \, dx\). So, set up the integral for the area as \(\int_{0}^{2} \frac{x^{2}}{\sqrt{9 - x^{2}}} \, dx\).

Consider a substitution to simplify the integral. Since the denominator involves \(\sqrt{9 - x^{2}}\), a trigonometric substitution such as \(x = 3 \sin \theta\) is appropriate because it will simplify the square root expression.

Perform the substitution: express \(x^{2}\), \(dx\), and \(\sqrt{9 - x^{2}}\) in terms of \(\theta\). For example, \(x = 3 \sin \theta\) implies \(dx = 3 \cos \theta \, d\theta\) and \(\sqrt{9 - x^{2}} = 3 \cos \theta\). Also, update the limits of integration accordingly by substituting \(x=0\) and \(x=2\) into \(x=3 \sin \theta\) to find the new \(\theta\) limits.

Rewrite the integral in terms of \(\theta\), simplify the integrand, and then integrate with respect to \(\theta\). After integrating, substitute back to \(x\) if necessary to express the area in terms of the original variable.

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주요 개념

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

Definite Integral for Area Calculation

The definite integral of a function over an interval [a, b] represents the net area between the graph of the function and the x-axis. When the function is non-negative on the interval, the integral gives the exact area under the curve. This concept is essential for finding the area bounded by g(x) and the x-axis on [0, 2].

Integrating Functions with Radical Expressions

Functions involving radicals, such as g(x) = x² / √(9 - x²), often require substitution methods for integration. Recognizing the form and applying appropriate substitutions, like trigonometric substitution, simplifies the integral and makes it solvable. This technique is crucial for handling the integral of g(x).

Domain and Interval Considerations

Understanding the domain of the function and the interval of integration ensures the integral is properly set up. Since g(x) involves a square root in the denominator, the expression under the root must be positive, restricting the domain. Confirming the function is defined and continuous on [0, 2] is necessary before integrating.

Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.b. Find the area of the region bounded by the graph of g and the x-axis on the interval [0,2].