find the area of the regin bounded by the graphs of y=-x^2=2x=3 and y=3.
i don't need help solving the problem and but i am a little confused.
ok the graph is a parabola and i drew a parobla with y= 3. now when find the area, am I finding the area on the top, above y=3 or on the bottom, below y=3.
i hope my question makes sense. basically, whick part of the graph am I finding the area of. top or bottom?
sorry about the equation. it is actually y=-x^2+2x+3
find the area from the top of the parabola down to y=3
above ("top" of) y=3
y=3 cuts the parabola at (-2,3)and (0,3), so the area bounded is the area of the parabola below the line y=3.
its area is integral(3 - x^2 - 2x - 3)dx from -2 to 0
To find the area bounded by the graphs of y = -x^2 + 2x + 3 and y = 3, you are looking for the area below the parabola and above the line y = 3.
To calculate this area, you need to set up an integral that represents the area between the curves.
First, let's find the points of intersection between the parabola and the line y = 3. Set -x^2 + 2x + 3 = 3 and solve for x:
-x^2 + 2x + 3 = 3
-x^2 + 2x = 0
x(-x + 2) = 0
So, x = 0 or x = 2.
These are the x-coordinates of the points of intersection. To find the y-coordinates, substitute these x-values into the equation of the parabola:
For x = 0:
y = -(0^2) + 2(0) + 3
y = 3
For x = 2:
y = -(2^2) + 2(2) + 3
y = -4 + 4 + 3
y = 3
Therefore, the points of intersection are (0, 3) and (2, 3).
Now, you can set up the integral to find the area:
The area is given by the integral of the difference between the curves:
∫ [3 - (-x^2 + 2x + 3)] dx from 0 to 2.
Simplifying the integral:
∫ [3 + x^2 - 2x - 3] dx from 0 to 2
∫ [x^2 - 2x] dx from 0 to 2
To evaluate this integral, you need to integrate term by term:
∫ x^2 dx - ∫ 2x dx from 0 to 2
Integrating, you get:
(x^3/3) - (x^2) from 0 to 2 - 2(x^2/2) from 0 to 2
(8/3) - 4 - (2)(0/2) - 0
8/3 - 4
8/3 - 12/3
-4/3
So, the area bounded by the graphs of y = -x^2 + 2x + 3 and y = 3, above y = 3, is -4/3.
Therefore, the area on the "top" of the parabola above y = 3 is -4/3 (or approximately -1.33).