Find the area of the region bounded by the graphs of y = x, y = 6 - 2x, and y = 0. (4 points)


3
6
9
None of these

Please help I don't know how to do this

intersection of y = x and y = 6-2x

x = 6-2x
3x = 6
x = 2 , so y = 2

so you have a triangle with vertices
(0,0), (2,2) and ((3,0), the last one is the x-intercept.

Surely you don't Calculus to do this.
base = 3, height = 2
Area = (1/2)(3)(2) = 3

However, if you want to exercise your calculus, you can do it as

vertical strips:

∫[0,2] x dx + ∫[2,3] 6-2x dx = 2+1 = 3

or as horizontal strips:
∫[0,2] (3 - y/2)-y dy = 3

To find the area of the region bounded by the graphs, we need to first determine the points where the graphs intersect.

Setting y = x and y = 6 - 2x equal to each other, we have:

x = 6 - 2x

3x = 6

x = 2

Substituting this value of x back into either equation, we can find the corresponding y-values:

For y = x, when x = 2, y = 2.

For y = 6 - 2x, when x = 2, y = 6 - 2(2) = 2.

So, the points of intersection are (2, 2) and (2, 2).

Next, we can sketch the three equations to visualize the region bounded by the graphs:

| /\
6 | /
| /
5 |/
------------
0 1 2 3

The region is a triangle with base 2 and height 2.

To find the area of a triangle, we use the formula: A = 1/2 * base * height.

Plugging in the values, we have:

A = 1/2 * 2 * 2

A = 2

Therefore, the area of the region bounded by the graphs is 2.

Therefore, the correct answer is None of these.

To find the area of the region bounded by the graphs of those three equations, we need to find the points of intersection first. The three equations are y = x, y = 6 - 2x, and y = 0.

Setting y = x equal to y = 6 - 2x, we can solve for x:
x = 6 - 2x
3x = 6
x = 2

So one of the points of intersection is (2, 2).

Next, setting y = x equal to y = 0, we can solve for x:
x = 0

So another point of intersection is (0, 0).

Finally, setting y = 6 - 2x equal to y = 0, we can solve for x:
6 - 2x = 0
2x = 6
x = 3

So the last point of intersection is (3, 0).

Now that we have the points of intersection (0, 0), (2, 2), and (3, 0), we can find the area of the bounded region.

We have two triangles and a rectangle. The area of a triangle is given by the formula A = (1/2) * base * height, and the area of a rectangle is given by the formula A = length * width.

First, consider the triangle with base (0 to 2) and height (0 to 2). The area of this triangle is (1/2) * 2 * 2 = 2.

Next, consider the triangle with base (2 to 3) and height (2 to 0). The area of this triangle is (1/2) * (3 - 2) * 2 = 1.

Finally, consider the rectangle with length (0 to 3) and width 2. The area of this rectangle is 3 * 2 = 6.

Adding up the areas of the two triangles and the rectangle, the total area of the bounded region is 2 + 1 + 6 = 9.

Therefore, the answer is 9.