# Find the area between the curve # y = e^(2x) # y = 0 from x = 1 to 3 ?

Area

The shaded are sought is given by the integral:

# A = int_1^3 \ e^(2x) \ dx #

# \ \ \ = [ \ e^(2x)/2 \ ]_1^3 #

# \ \ \ = 1/2 (e^6-e^2) #

# \ \ \ = 198.019868 ... #

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To find the area between the curve ( y = e^{2x} ) and the x-axis from ( x = 1 ) to ( x = 3 ), you need to compute the definite integral of ( e^{2x} ) with respect to ( x ) over the interval ([1, 3]):

[ \text{Area} = \int_{1}^{3} e^{2x} , dx ]

To integrate ( e^{2x} ), use the following antiderivative:

[ \int e^{ax} , dx = \frac{1}{a} e^{ax} + C ]

where ( a ) is a constant. In this case, ( a = 2 ). So,

[ \int e^{2x} , dx = \frac{1}{2} e^{2x} + C ]

Evaluate the definite integral:

[ \text{Area} = \left[\frac{1}{2} e^{2x}\right]_{1}^{3} ] [ = \left(\frac{1}{2} e^{6}\right) - \left(\frac{1}{2} e^{2}\right) ] [ = \frac{1}{2} (e^{6} - e^{2}) ]

Therefore, the area between the curve ( y = e^{2x} ) and the x-axis from ( x = 1 ) to ( x = 3 ) is ( \frac{1}{2} (e^{6} - e^{2}) ).

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To find the area between the curve y = e^(2x) and the x-axis from x = 1 to x = 3, you need to integrate the absolute value of the function y = e^(2x) with respect to x over the interval [1, 3]. This is because the function e^(2x) is always positive for real values of x, so the area between the curve and the x-axis will be the integral of e^(2x) from x = 1 to x = 3.

∫[1 to 3] |e^(2x)| dx = ∫[1 to 3] e^(2x) dx

To integrate e^(2x), you can use the substitution method. Let u = 2x, then du = 2 dx.

∫ e^(2x) dx = (1/2) ∫ e^u du = (1/2) e^u + C

Now substitute back for u:

= (1/2) e^(2x) + C

Evaluate this expression from x = 1 to x = 3:

= (1/2) [e^(6) - e^(2)]

So, the area between the curve y = e^(2x) and the x-axis from x = 1 to x = 3 is (1/2) [e^(6) - e^(2)].

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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