# Find the region enclosed by the curves: y=e^(x) , y=xe^(x) , x=0

You need to find the limits of integration, hence, you need to solve the equation e^x = x*e^x , such that:

e^x = x*e^x => e^x - x*e^x = 0

Factoring out e^x yields:

e^x(1 - x) = 0

Since e^x > 0 , hence 1 - x = 0...

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You need to find the limits of integration, hence, you need to solve the equation e^x = x*e^x , such that:

e^x = x*e^x => e^x - x*e^x = 0

Factoring out e^x yields:

e^x(1 - x) = 0

Since e^x > 0 , hence 1 - x = 0 => x = 1

You need to evaluate the functions y = e^x and y = x*e^x at x = 0 and x = 1/2 , such that:

x = 0 => y = e^0 => y = 1

x = 0 => y = 0*e^0 = 0

x = 1/2 => y = e^(1/2) => y = sqrt e 

x = 1/2 => y = (sqrt e)/2

Hence e^x > x*e^x  for x in [0,1] , thus, evaluating the area of the region enclosed by the given curves, yields:

int_0^1 (e^x - x*e^x)dx

Using the property of linearity of integral, you need to split the integral in two, such that:

int_0^1 (e^x)dx - int_0^1 (x*e^x)dx

You need to solve int_0^1 (x*e^x)dx using integration by parts, such that:

int udv = uv - int vdu

u = x => du = dx

dv = e^x => v = e^x

int_0^1 (x*e^x)dx= x*e^x|_0^1 - int_0^1 e^x dx

int_0^1 (e^x - x*e^x)dx = int_0^1 (e^x)dx -x*e^x|_0^1 + int_0^1 e^x dx

int_0^1 (e^x - x*e^x)dx = 2e^x|_0^1 - x*e^x|_0^1

Using the fundamental theorem of calculus, yields:

int_0^1 (e^x - x*e^x)dx = 2e^1 - 2e^0 - 1*e^1 + 0*e^0

int_0^1 (e^x - x*e^x)dx = 2e - 2 - e

int_0^1 (e^x - x*e^x)dx = e - 2

Hence, evaluating the area of the region enclosed by the given curves, yields int_0^1 (e^x - x*e^x)dx = e - 2.`

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