find the are bounded by the x-axis,y-axis and the curve x^2+y^2+16x+16y+64=0.
The area is bounded by 1/4 of the circle with center at (-8,8) and radius 8. So, it's just
r^2 - 1/4 (pi r^2) 64-(1/4)pi*8^2 = 64-16pi
To find the area bounded by the x-axis, y-axis, and the curve x^2 + y^2 + 16x + 16y + 64 = 0, we can follow these steps:
Step 1: Rewrite the equation in standard form by completing the square. This will help us identify the center and radius of the circle.
x^2 + y^2 + 16x + 16y + 64 = 0
Rearrange the equation by grouping the terms and completing the square for x and y:
(x^2 + 16x) + (y^2 + 16y) + 64 = 0
To complete the square for x, add (16/2)^2 = 64 to both sides. Similarly, to complete the square for y, add (16/2)^2 = 64 to both sides:
(x^2 + 16x + 64) + (y^2 + 16y + 64) + 64 = 64 + 64 + 64
Now, we can factor the squares:
(x + 8)^2 + (y + 8)^2 + 64 = 192
Simplify:
(x + 8)^2 + (y + 8)^2 = 128
Based on this equation, we can determine that the center of the circle is (-8, -8) and the radius is sqrt(128) = 8sqrt(2).
Step 2: Sketch the graph of the equation to visualize the given curve:
Since the equation represents a circle with a center at (-8, -8) and a radius of 8sqrt(2), you can draw a circle with these specifications on a coordinate plane.
Step 3: Identify the points of intersection of the circle with the x-axis and y-axis:
To find the points of intersection with the x-axis, we set y = 0 in the equation:
(x + 8)^2 + (0 + 8)^2 = 128
(x + 8)^2 + 64 = 128
(x + 8)^2 = 128 - 64 = 64
(x + 8)^2 = 64
x + 8 = ±sqrt(64)
x + 8 = ±8
x = -8 ± 8
So, the x-axis intersection points are (-16, 0) and (0, 0).
To find the points of intersection with the y-axis, we set x = 0 in the equation:
(0 + 8)^2 + (y + 8)^2 = 128
64 + (y + 8)^2 = 128
(y + 8)^2 = 128 - 64 = 64
(y + 8)^2 = 64
y + 8 = ±sqrt(64)
y + 8 = ±8
y = -8 ± 8
So, the y-axis intersection points are (0, -16) and (0, 0).
Step 4: Calculate the area of the region bounded by the x-axis, y-axis, and the curve:
To calculate the area, we need to determine if the circle lies inside or partially outside the quadrant defined by the x-axis and y-axis.
Based on the graph, we can see that the circle intersects both the x-axis and y-axis within the positive quadrant, creating a sector.
The area of the sector can be calculated using the formula:
Area of sector = (1/2) * r^2 * θ
Since the radius is 8sqrt(2), the area of the sector is:
Area of sector = (1/2) * (8sqrt(2))^2 * (π/2) = 256π
To find the area bounded by the x-axis, y-axis, and the curve, subtract the area of the sector from the area of the rectangular region defined by the x-axis and y-axis:
Area bounded = (16 * 16) - 256π
Therefore, the area bounded by the x-axis, y-axis, and the curve x^2 + y^2 + 16x + 16y + 64 = 0 is 256π - 256 square units.