Final answer:
The area under the curve y = 2 √ (36 − 4x^2) from x = 0 to x = 2 is found by integrating the function. It represents a quarter circle with radius 6, and the area is 2 times the area of a quarter circle. The exact or approximate numerical answer depends on whether we express it in terms of π or as a decimal approximation.
Step-by-step explanation:
The area bounded by the curve y = 2 √ (36 − 4x2), the x-axis, and the lines x = 0 and x = 2 can be found using an integral. Since the function involves a square root and a square, it reminds us of the equation of a circle, which suggests that the curve is part of a circle, and the bounded area will be a sector of that circle. To find the area under the curve from x = 0 to x = 2, we need to integrate the function y = 2 √ (36 − 4x2) with respect to x over this interval. Since the function represents the upper half of a circle with radius 6, which is twice the length of y when solving for y given x, the area can be calculated as:
∫02 2 √ (36 − 4x2) dx
To solve the integral, we can use a trigonometric substitution or recognize that the integrand represents the area element of a semicircle. Given the symmetry and the radius, this integral simplifies to calculating the area of a quarter circle of radius 6 and then doubling it, because the integrand has already been multiplied by 2. Hence, the area is:
Area = 2 * (π * 62)/4 = 18π
The exact numeric answer depends on whether the options provided were expected to be in terms of π or not; if the options were meant to be in decimals, we would use π ≈ 3.14 to find the approximate numerical value.