Find the area under the curve y= x+ 6/x, x=1 to x=2.
The area under the curve y= x+ 6/x and x axis is calculated using Leibniz-Newton formula.
Int ( x+ 6/x)dx = F(b) - F(a), where a = 1 and b = 2
First, we'll determine the result of the indefinite integral:
Int ( x+ 6/x)dx
We'll use the additive property of integrals:
Int ( x+ 6/x)dx = Int xdx + Int 6dx/x
We'll simplify and we'll take out the constants and we'll get:
Int ( x+ 6/x)dx = Int xdx + 6Int dx/x
Int ( x+ 6/x)dx = x^2/2 + 6ln |x| + C
The resulted expression is F(x).
Now, we'll determine F(b) = F(2):
F(2) = 6ln |2| + 2^2/2
F(2) = 6ln |2| + 2
Now, we'll determine F(a) = F(1):
F(1) = 6ln |1| + 1^2/2
F(1) = 0 + 1/2
We'll determine the area:
A = F(2) - F(1)
A = 6ln |2| + 2 - 1/2
A = ln 2^6 + 3/2
A = ln 64 + 1.5 square units
A = 4.15 + 1.5
A = 5.65 square units