Calculate the definite integral of f(x)=e^2x from x=0 to x=2.
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f(x) = e^2x
Let F(x) = intg f(x)
==> F(x) = intg e^2x dx
= e^2x/2 + c
==> F(x) = 2^2x/2 + C
Now the difinite integral from x= 0 to x= 2 is the area between the f(x) and x= 0 and x= 2:
A = F(2) - F(0)
=(e^2*2/2 + c ) - (e^2*0 / 2 + c)
= (e^4)/2 - e^0/2
= (e^4)/2 - 1/2
= (e^4 - 1)/2
==> The definite integral is (e^4 - 1)/2
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The definite integral is the area which has to be found, that is located between the given curve y = e^2x and the lines x = 0 and x = 2, also the x axis.
To calculate the area, we'll use the formula:
S = Integral (f(x) - ox)dx = Int f(x)dx = Int e^(2x) dx
Int e^(2x) dx = e^(2x)/2 + C
Now, we'll calculate the value of the area, using Leibnitz Newton formula::
S = F(2) - F(0), where
F(2) = e^(2*2)/2 = e^4/2
F(0) = e^(2*0)/2 = e^0/2 = 1/2
S = e^4/2 - 1/2
S = (e^4 - 1)/2
We have a difference of squares, at numerator:
S = (e^2-1)(e^2+1)/2
S = (e-1)(e+1)(e^2+1)/2
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