Final answer:
The area bounded by the x-axis, the y-axis, and the curve y=cos⁽⁻¹⁾(9x) is 0.243 square units. Option A is correct.
Step-by-step explanation:
To calculate the area bounded by the x-axis, the y-axis, and the curve y=cos⁽⁻¹⁾(9x), we need to find the points where the curve intersects the x-axis. Set y = 0 and solve for x:
0 = cos⁽⁻¹⁾(9x)
1 = 9x
x = 1/9
So the curve intersects the x-axis at x = 1/9. Now we can find the area by integrating the curve from 0 to 1/9:
A = ∫[0, 1/9] cos⁽⁻¹⁾(9x) dx
Using the substitution u = 9x, du = 9 dx, the integral becomes:
A = (1/9) ∫[0, 1] cos⁽⁻¹⁾(u) du
Using the formula for the integral of the inverse cosine function, A = (1/9) (u cos⁽⁻¹⁾(u) + √(1 - u²) + C), where C is the constant of integration.
Substituting u = 9x and evaluating the integral, we get:
A = (1/9) ((9x) cos⁽⁻¹⁾(9x) + √(1 - (9x)²) + C)
Plugging in the limits of integration, A = (1/9) * (1/9) * (9/√80)
A = 1/720 √80 = 0.243 square units.