Answer to Question #265103 in Calculus for seed

Question #265103
Find the area of the region enclosed by the graphs of y= 3/x and y=4-x
1
Expert's answer
2021-11-15T07:35:48-0500

Given: y=3/x, y=4xy=3/x ,\ y=4-x

Point of intersection of these two curve will occur at

3/x=4xx24x+3=0(x3)(x1)=0x=1,33/x=4-x\Rightarrow x^2-4x+3=0 \\\Rightarrow (x-3)(x-1)=0 \\\Rightarrow x=1,3

Graph of the above function is shown below




Area of the region enclosed by these two curves

A=13(4x3x)dx=[4xx223lnx]13=12923ln34+12+3ln1=43ln3A=\int_{1}^{3} (4-x-\frac{3}{x})dx=[4x-\frac{x^2}{2}-3lnx]_{1}^{3}\\ =12-\frac{9}{2}-3ln3-4+\frac{1}{2}+3ln1=4-3ln3 [ln1=0\because ln 1=0 ]


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