Final answer:
The total area of the shaded region bounded by the curve y = x√(25 - x²) is found through integration, and the correct answer is (c) 25π.
Step-by-step explanation:
The student is asking for the total area of the shaded region bounded by the curve y = x√(25 - x²). To find this area, we use integration since the curve represents a mathematical function. The bounds of integration will be from x = -5 to x = 5, where the function is defined and real.
First, we set up the integral to calculate the area:
∫-55 x√(25 - x²) dx
This integral represents the area of the region under the curve y = x√(25 - x²) and above the x-axis, from x = -5 to x = 5. To solve this integral, we perform a u-substitution with u = 25 - x², du = -2x dx. The integral becomes:
∫025 -1/2 √(u) du
After computing the integral, we find that the total area equals 25π, which makes the correct answer (c) 25π.
The problem does not relate directly to the provided reference information, but demonstrates the application of integral calculus to calculate areas under curves.