How do you find the area between #g(x)=4/(2-x), y=4, x=0#?

Answer 1

The area is #=4-4ln2#

A small area is given by #dA=(4-4/(2-x))dx#
The point of intersection of #g(x)# and #y=4#
is #4=4/(2-x)# #=>#, #2-x=1# #=>##x=1#

Therefore, the area is

#A=int_0^1(4-4/(2-x))dx=[4x+4ln(2-x)]_0^1 #
#=4+0-0-4ln2=4-4ln2#

graph{(y-4/(2-x))(y-4)=0 [-7.1, 6.95, -0.03, 6.99]}

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Answer 2

To find the area between the curves ( g(x) = \frac{4}{2 - x} ), ( y = 4 ), and ( x = 0 ), you can follow these steps:

  1. Find the points of intersection between the curves ( g(x) ) and ( y = 4 ) by setting them equal to each other and solving for ( x ).
  2. Determine the interval over which ( g(x) ) is bounded by ( y = 4 ).
  3. Integrate ( g(x) - 4 ) over the interval found in step 2 to calculate the area between the curves.

Let's go through these steps:

  1. ( g(x) = \frac{4}{2 - x} ) intersects ( y = 4 ) when ( \frac{4}{2 - x} = 4 ). Solving this equation gives ( x = 1 ).

  2. The interval over which ( g(x) ) is bounded by ( y = 4 ) is from ( x = 0 ) to ( x = 1 ).

  3. Integrate ( g(x) - 4 ) from ( x = 0 ) to ( x = 1 ): [ \int_{0}^{1} \left( \frac{4}{2 - x} - 4 \right) dx ] [ = \int_{0}^{1} \left( \frac{4}{2 - x} - \frac{4(2 - x)}{2 - x} \right) dx ] [ = \int_{0}^{1} \frac{4 - 4(2 - x)}{2 - x} dx ] [ = \int_{0}^{1} \frac{4 - 8 + 4x}{2 - x} dx ] [ = \int_{0}^{1} \frac{4x - 4}{2 - x} dx ] [ = \int_{0}^{1} \frac{4(x - 1)}{x - 2} dx ]

You can then proceed to evaluate this integral to find the area between the curves.

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Answer from HIX Tutor

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.

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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