Evaluate the integral by interpreting it in terms of areas.
The integral from -3 to 0 of 5+(9-x^2)^1/2.
It's just a 3x5 rectangle + 1/4 circle of radius 3.
To evaluate the given integral by interpreting it in terms of areas, we can think of the integral as finding the area under the curve of the function f(x) = 5 + √(9 - x^2) between x = -3 and x = 0.
First, let's analyze the function f(x) = 5 + √(9 - x^2). The function represents a semi-circle with center (0, 0) and a radius of 3. The 5 added to the function represents a translation upwards by 5 units.
Now, we'll split the integral into two parts to calculate the areas separately. We'll have one part for the positive values of the function and one part for the negative values.
For positive values:
The integral from -3 to 0 of f(x) represents the area between the x-axis and the curve. We can think of this as the area of the semi-circle with radius 3, subtracting the area of the triangle formed by the x-axis and the points (0, 0) and (0, 5).
The area of the semi-circle is (1/2)π(3^2) = (9/2)π.
The area of the triangle is (1/2)(3)(5) = 15/2.
So, the area for the positive values is (9/2)π - 15/2.
For negative values:
The integral from -3 to 0 of -f(x) represents the area between the x-axis and the curve but below the x-axis. Since the curve is symmetric about the x-axis, this area will be equal to the area for the positive values.
Therefore, the area for the negative values is also (9/2)π - 15/2.
To find the total area, we can add the areas for positive and negative values:
Total area = (9/2)π - 15/2 + (9/2)π - 15/2
= (18/2)π - 30/2
= 9π - 15.
Hence, the integral from -3 to 0 of 5 + √(9 - x^2) is equal to 9π - 15.