Find the área of the región bounded by the graphs of y=x, y=-x+4, and y=0

A) 4
B) 2
C) 8
D) no me of these

using the symmetry of the region

A = 2∫[0,2] (4-x)-x dx
Now just plug and chug

check: you have two triangles, each of base 4 and height 2.

Could you explain me more, pleaseee

what - the concept of an integral and areas, or just how to evaluate the integral? It's a simple application of the power rule.

Divide the area up into thin strips, each of width dx, and with height the distance between the lines.

∫ (4-x)-x dx = ∫4-2x dx = 4x - x^2
Now evaluate at the limits and subtract.

Still confused? Read your text -- I'm sure it has examples.

Thanks Oobleck

To find the area of the region bounded by the graphs of y=x, y=-x+4, and y=0, we can use basic calculus and integration.

1. First, let's graph the given functions to visualize the region.

The graph of y=x is a straight line passing through the origin and with a slope of 1.

The graph of y=-x+4 is also a straight line passing through the point (0,4) and with a slope of -1.

The graph of y=0 is the x-axis.

So, we have a triangle formed by these three lines.

2. To find the area of the region, we need to find the x-values at which the two lines intersect.

Setting y=x and y=-x+4 equal to each other, we find:

x = -x + 4

2x = 4

x = 2

So, the lines intersect at x = 2.

3. Now, we can set up the integral to find the area. Since the region is bounded by the lines y=x and y=-x+4, we need to integrate the difference in their equations over the interval [0, 2] to find the area.

For the upper curve, y=-x+4, the integral is: ∫[-x+4]dx from 0 to 2.

For the lower curve, y=x, the integral is: ∫[x]dx from 0 to 2.

4. Evaluating the integral, we get:

Upper curve: ∫[-x+4]dx = -[(1/2)x^2 - 4x] from 0 to 2

Lower curve: ∫[x]dx = (1/2)x^2 from 0 to 2

Substituting the limits of integration, we have:

Upper curve: [-((1/2)(2)^2 - 4(2))] - [-((1/2)(0)^2 - 4(0))]

= -(2 - 8) - -(0 - 0)

= -(-6) = 6

Lower curve: (1/2)(2)^2 - (1/2)(0)^2

= (1/2)(4) - (1/2)(0)

= 2 - 0 = 2

5. Finally, we find the difference between the upper curve and the lower curve to get the area:

Area = 6 - 2 = 4

Therefore, the area of the region bounded by the graphs of y=x, y=-x+4, and y=0 is 4.

Therefore, the correct option is A) 4.