Final answer:
To evaluate the integral ∫_-3^0(5+√(9-x²)) d x, we can interpret it in terms of areas. The integral represents the area under the curve of the function f(x) = 5+√(9-x²) from x = -3 to x = 0. We can break this area into two parts: the area under the curve of f(x) = 5+√(9-x²) from x = -3 to x = -2, and the area under the curve of f(x) = 5+√(9-x²) from x = -2 to x = 0.
Step-by-step explanation:
To evaluate the integral ∫-30(5+√(9-x²)) dx, we can interpret it in terms of areas. The integral represents the area under the curve of the function f(x) = 5+√(9-x²) from x = -3 to x = 0. We can break this area into two parts: the area under the curve of the function f(x) = 5+√(9-x²) from x = -3 to x = -2, and the area under the curve of the function f(x) = 5+√(9-x²) from x = -2 to x = 0.
To find the area under the curve of f(x) = 5+√(9-x²) from x = -3 to x = -2, we can use the formula for the area of a trapezoid: A = (b₁ + b₂)h/2. In this case, b₁ is the function value at x = -3, b₂ is the function value at x = -2, and h is the difference between -3 and -2.
To find the area under the curve of f(x) = 5+√(9-x²) from x = -2 to x = 0, we can use the formula for the area of a semicircle: A = πr²/2. In this case, the radius r is the function value at x = 0.