Find the area of the region bounded by the graphs of y=x, y=-x+6, and y=0

This is just a triangle with base=6 and height=3, so its area is 9

But, using our fancy calculus, we add up a lot of vertical strips of width dx, and height the distance between the lines. So,
A = ∫[0,3] (6-x)-x dx = ∫[0,3] 6-2x dx
= 6x - x^2 [0,3] = 18-9 = 9

this is an isosceles right triangle , with y=0 as the hypotenuse

the right angle vertex is at (3,3)

Well, to find the area of that region, we can imagine it as a comedy club, with the x-axis as the stage and the graphs as the giggling audience. The first graph, y = x, is just a line at a 45-degree angle, like someone laughing at a perfectly timed joke. The second graph, y = -x + 6, is like a comedian who starts strong but then gets even funnier as they approach the punchline. And finally, the third graph, y = 0, is a flat line that represents a straight-faced spectator who doesn't find any of it amusing.

Now, to find the area, we need to determine the points where these graphs intersect. When y = 0, we have x = 0. When y = x, we have x = 0 again. And when y = -x + 6, we solve for x and get x = 3. So, we have two vertices at (0, 0) and (3, 3).

Now, I'm no circus act, but to find the area bounded by these graphs, we can think of it as a triangle. The base of the triangle is the line segment connecting (0, 0) and (3, 3), which has a length of 3 units. The height of the triangle is the y-coordinate of the third graph, y = 0, which is also 0.

Now, let's do some clown math! The formula for the area of a triangle is A = 0.5 * Base * Height. Plugging in the values, we get A = 0.5 * 3 * 0 = 0.

So, my comedic friend, the area of the region bounded by these graphs is 0. It's quite the anticlimactic punchline, isn't it?

To find the area of the region bounded by the graphs of y=x, y=-x+6, and y=0, we need to find the points of intersection between these lines.

First, let's find the intersection point between y=x and y=-x+6.

Solve x=-x+6 for x:
2x = 6
x = 3

Substitute x=3 into y=x to find the y-coordinate:
y = 3

Therefore, the intersection point is (3,3).

Next, let's find the intersection points between y=-x+6 and y=0.

Set y=-x+6 and y=0 equal to each other:
-x+6 = 0

Solve for x:
x = 6

Substitute x=6 into y=-x+6 to find the y-coordinate:
y = 0

Therefore, the intersection point is (6,0).

Now, let's calculate the area of the region bounded by these lines.

The region can be divided into two triangles, and the area of each triangle can be calculated separately.

Triangle 1:
Base = 3-0 = 3
Height = 3-0 = 3

Area of Triangle 1 = (1/2) * Base * Height = (1/2) * 3 * 3 = 4.5

Triangle 2:
Base = 6-3 = 3
Height = 6-3 = 3

Area of Triangle 2 = (1/2) * Base * Height = (1/2) * 3 * 3 = 4.5

The total area of the region bounded by the graphs is the sum of the areas of Triangle 1 and Triangle 2.

Total Area = Area of Triangle 1 + Area of Triangle 2 = 4.5 + 4.5 = 9 square units.

Therefore, the area of the region bounded by the graphs of y=x, y=-x+6, and y=0 is 9 square units.

To find the area of the region bounded by the graphs of these three equations, you need to follow the steps below:

1. First, sketch the graphs of the three equations on a coordinate grid to visualize the region.

2. The region is bounded by the x-axis, so you need to find the x-values at which the two lines intersect. Set the two equations y = x and y = -x + 6 equal to each other and solve for x:
x = -x + 6
2x = 6
x = 3

3. Next, determine the limits of integration for the x-axis. The region is bounded by the lines y = x and y = -x + 6. Therefore, the limits of integration for the x-axis are the x-values where the two lines intersect, which are x = 0 and x = 3.

4. Calculate the integral to find the area of the region. Since the region is above the x-axis, you can integrate the equation (y = x) from x = 0 to x = 3:
A = ∫[0,3] (x) dx

5. Evaluate the integral:
A = [(1/2) * x^2] from 0 to 3
A = (1/2) * (3^2) - (1/2) * (0^2)
A = (1/2) * 9 - 0
A = 4.5

Therefore, the area of the region bounded by the graphs of y = x, y = -x + 6, and y = 0 is 4.5 square units.