# How do you find the area between #f(x)=10/x, x=0, y=2, y=10#?

Area

First examine the area drawn on a graph:

The bounded area,

# A_1 = 1 xx 8 = 8 #

And the remaining Area, is that under the curve

#10/x=2 => x=5# , and#10/x=10=>x=1#

This area is then given by:

# A_2 = int_1^5 (10/x-2) \ dx #

# \ \ \ \ \= [10lnx-2x color(white)int]_1^5#

# \ \ \ \ \= (10ln5-10) - (10ln1-2)#

# \ \ \ \ \= 10ln5-10 - 0 + 2#

# \ \ \ \ \= 10ln5-8#

And so the total bounded area is:

#A= A_1+ A_2#

#\ \ \ = 8+10ln5-8#

#\ \ \ = 10ln5#

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graph{x(y-2)(y-10)(xy-10)=0 [-20, 20, -10, 10]}

The curved boundary is the branch of the rectangular hyperbola

The area is

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To find the area between the curve ( f(x) = \frac{10}{x} ) and the lines ( x = 0 ), ( y = 2 ), and ( y = 10 ), you need to set up a definite integral. First, determine the points of intersection of the curve and the lines. Then integrate the absolute difference between the upper and lower functions with respect to ( x ) over the interval of intersection. The integral setup is as follows:

[ \int_{a}^{b} \left| f(x) - g(x) \right| , dx ]

where ( f(x) = \frac{10}{x} ), ( g(x) ) represents the appropriate line, and ( a ) and ( b ) are the ( x )-values of the points of intersection.

Find the points of intersection by setting ( f(x) ) equal to the lines ( y = 2 ) and ( y = 10 ), then solve for ( x ). These points will be your limits of integration. After finding the intersection points, integrate the absolute difference between ( f(x) ) and the lines from the lower limit to the upper limit. This will give you the area between the curve and the specified lines.

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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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