Find the points of intersection of the graphs of the functions y=4x^2-25 and y=0. -5/2,5/2
Find the area bounded by the curves y=4x^2-25, y=0, x=-1, x=3.5. PS the area is not 54 I already tried.
Your intersection points are correct.
The problem is that the second boundary value of x = 3.5 is beyond the intersection point of (2.5,0), and you have a part of the region that lies above the x-axis. So you have to find the area in two parts:
from x = -1 to x=2.5, and then from x = 2.5 to 3.5
area = ∫(25-4x^2) dx from -1 to 2.5 + ∫(4x^2-25)dx from 2.5 to 3.5
= [25x - (4/3)x^3] from -1 to 2.5 + [(4/3)x^3 - 25x] from 2.5 to 3.5
= 62.5 - 125/6 - (-25 + 4/3) + 81/2 - 87.5 - (125/6 - 62.5)
= 60
check my arithmetic
To find the points of intersection between the graphs of the functions y=4x^2-25 and y=0, we need to set the two equations equal to each other and solve for x.
Setting y=4x^2-25 equal to y=0, we get:
4x^2-25 = 0
To solve this quadratic equation, we can factor it:
(2x+5)(2x-5) = 0
Setting each factor equal to zero, we have:
2x+5=0 or 2x-5=0
Solving these equations, we find:
2x = -5 or 2x = 5
x = -5/2 or x = 5/2
Therefore, the points of intersection between the two graphs are (-5/2, 0) and (5/2, 0).
Now, to find the area bounded by the curves y=4x^2-25, y=0, x=-1, and x=3.5, we can calculate the definite integral of the difference between the two functions within the given limits.
The area can be found using the formula:
Area = ∫[a, b] (f(x) - g(x)) dx
In this case:
f(x) = 4x^2-25
g(x) = 0
a = -1
b = 3.5
The integral we need to evaluate is:
Area = ∫[-1, 3.5] (4x^2-25) dx
To find this area, you can evaluate the integral using anti-derivatives or find the area under the curve using numerical methods like numerical integration or approximation techniques.
If you have tried numerical methods and obtained an incorrect result of 54, it is likely that there was an error in the approximation or calculation. Double-check your calculations or consider using a different numerical method to obtain a more accurate result.