What is the area between f(x)=5x^4+3x^2, x=1 and x=2 ?
We notice that it is not established the other boundary curve, we'll suppose that we have to calculate the area between f(x), the lines x=1 and x=2 and the x axis.
The definite integral will be calculated with Leibniz-Newton formula:
Int f(x)dx = F(b)-F(a)
We'll calculate the indefinite integral of f(x):
Int f(x)dx = Int (5x^4+3x^2)dx
We'll use the property of integral to be additive:
Int (5x^4+3x^2)dx = Int 5x^4dx + Int 3x^2dx
Int 5x^4dx = 5x^5/5 + C
Int 5x^4dx = x^5 + C
Int 3x^2dx = 3*x^3/3 + C
Int 3x^2dx =x^3 + C
Int (5x^4+3x^2)dx = x^5 + x^3 + C
F(2) - F(1) = 2^5 + 2^3 - 1^5 - 1^3
F(2) - F(1) = 32 + 8 - 2
F(2) - F(1) = 38
The area bounded by the curve of f(x) and the lines x=1, x=2 and x axis is A=38 square units.