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.