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

Question

Isaiah Allen · Accepted Answer

Area #= 10ln5#

First examine the area drawn on a graph:

The bounded area, #A# is made up of two components, the first is a rectangle bounded by #y=2#, #y=10#, #x=0# and #x=1#, so it's area is given by:

# A_1 = 1 xx 8 = 8 #

And the remaining Area, is that under the curve #y=10/x#, above #y=2# between #x=1# and #x=5#. I found these values graphically, but algebra will also provide the values:

#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#

Isabella Ackerman · Answer

#10 ln 5 = 16.1# areal units, nearly

graph{x(y-2)(y-10)(xy-10)=0 [-20, 20, -10, 10]} The curved boundary is the branch of the rectangular hyperbola xy = 10, in #Q_1#. See the graph. The area is #int x dy#, with x =10/y and y from 0 to 10. #=10 int 1/y dy#, y from 2 to 10 #=10[ln y],# between y = 2 and y = 10 #=10[ln10-ln2]# #=10ln(10/2)# #=10 ln 5# areal units.

Adam Adkins · Answer

undefined 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.