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.
c. Which region has greater area?

Verified step by step guidance

1. First, identify the two functions given: \(f(x) = \frac{x^{2}}{3}\) and \(g(x) = \frac{x^{2}}{\sqrt{9 - x^{2}}}\), both defined on the interval \([0, 2]\).
2. To compare the areas under the curves of \(f(x)\) and \(g(x)\) on \([0, 2]\), set up the definite integrals for each function: \(A_f = \int_0^2 \frac{x^{2}}{3} \, dx\) and \(A_g = \int_0^2 \frac{x^{2}}{\sqrt{9 - x^{2}}} \, dx\).
3. Evaluate each integral separately by applying appropriate integration techniques: for \(A_f\), use the power rule for integration; for \(A_g\), consider a trigonometric substitution such as \(x = 3 \sin \theta\) to simplify the integral.
4. After finding the expressions for both \(A_f\) and \(A_g\), compare their values to determine which area is greater.
5. Conclude which region has the greater area based on the comparison of the two definite integrals.

Key Concepts

Definite Integral as Area Under a Curve

The definite integral of a function over an interval represents the net area between the graph of the function and the x-axis. To find the area of a region bounded by a curve and the x-axis, we compute the integral of the function over the given interval.

Comparing Areas of Two Functions

To determine which region has a greater area between two functions on the same interval, calculate the definite integrals of each function separately over that interval. The function with the larger integral value corresponds to the region with the greater area.

Handling Functions with Domain Restrictions

When dealing with functions like g(x) = x²(9−x²)^(-1/2), it is important to consider the domain where the function is defined and real-valued. Ensuring the function is integrable on the interval [0,2] is essential before computing the area.