Let f(x)=x^3+3x and g(x=5x^2-x
Determine the area of the region enclosed by the graphs of f and g.
First, find where the graphs intersect:
x^3+3x = 5x^2-x
x^3 - 5x^2 + 4x = 0
x(x-4)(x-1) = 0
Now, f(x) > g(x) from 0 to 1
g(x) > f(x) from 1 to 4
If you want geometric area, break the integration into two intervals. If you want algebraic (signed) area, then just integrate from 0 to 4.
The actual integration is just powers, so it's simple.
To determine the area of the region enclosed by the graphs of functions f(x) and g(x), you need to find the points of intersection first. The area is then given by the definite integral of the difference between the two functions over the interval of intersection.
1. Find the points of intersection:
To find the points where the graphs of f(x) and g(x) intersect, set the two equations equal to each other and solve for x:
x^3 + 3x = 5x^2 - x
Simplifying the equation:
x^3 + 3x = 5x^2 - x
x^3 + 4x + x - 5x^2 = 0
x(x^2 + 4) + (x - 5x^2) = 0
x^3 + 4x + x - 5x^2 = 0
x^3 - 5x^2 + 5x + 4x = 0
x(x^2 - 5x + 4) + (x - 5x^2) = 0
Factoring out common terms:
x(x - 1)(x - 4) + (-x)(1 - 5x) = 0
(x - 1)(x)(x - 4) + (x)(5x - 1) = 0
(x - 1)(x)(x - 4 + 5x - 1) = 0
(x - 1)(x)(6x - 5) = 0
So, x = 1, x = 0, or x = 5/6.
2. Determine the limits of integration:
To calculate the area enclosed by the graphs of f and g, you need to determine the limits of integration. In this case, the x-values where the two graphs intersect are the limits of integration. So, the limits will be from x = 0 to x = 1, and from x = 0 to x = 5/6.
3. Calculate the area using the definite integral:
The area (A) is given by the definite integral of the difference between the two functions over the interval of integration:
A = ∫[0 to 1] (f(x) - g(x)) dx + ∫[0 to 5/6] (g(x) - f(x)) dx
By substituting the given functions f(x) and g(x) into the integral:
A = ∫[0 to 1] (x^3 + 3x - (5x^2 - x)) dx + ∫[0 to 5/6] ((5x^2 - x) - (x^3 + 3x)) dx
Evaluate the integrals:
A = ∫[0 to 1] (x^3 + 3x - 5x^2 + x) dx + ∫[0 to 5/6] (5x^2 - x - x^3 - 3x) dx
Simplify the expression inside each integral:
A = ∫[0 to 1] (-x^3 - 5x^2 + 4x) dx + ∫[0 to 5/6] (4x^2 - 4x^3 - 2x) dx
Evaluate the integrals using integral rules or techniques, such as the power rule.
Finally, calculate the definite integral to find the value of the enclosed area.