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Question

Sketch the graph of and evaluate

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Solution

The given equation is y=| x+3 | and integral to be evaluated is 6 0 | x+3 | dx.

To sketch the graph of the equation y=| x+3 |, put some values of x and find corresponding values of y. Make a table of values of y and x.

xy
63
52
41
30
21
12
03

Plot the points from the above table and join them by straight lines.



Figure (1)

Figure (1) shows the graph of the equation y=| x+3 |.

We have to evaluate the integral 6 0 | x+3 | dx. The absolute value function can be written as,

y=( x+3 ) 6x3 y=( x+3 ) 3x0

Thus, the integral can be written as,

6 0 | x+3 | dx= 6 3 ( x+3 ) dx+ 3 0 ( x+3 )dx

Integrate the above integral and apply the boundary conditions.

6 0 | x+3 | dx= 6 3 ( x+3 ) dx+ 3 0 ( x+3 ) dx = [ x 2 2 +3x ] 6 3 + [ x 2 2 +3x ] 3 0 =[ ( 3 ) 2 2 +3( 3 )( ( 6 ) 2 2 +3( 6 ) ) ]+[ ( 0 ) 2 2 +3( 0 )( ( 3 ) 2 2 +3( 3 ) ) ] =[ 9 2 918+18 ]+[ 9 2 +9 ]

Further, solve the above equation.

6 0 | x+3 | dx=[ 9 2 918+18 ]+[ 9 2 +9 ] =[ 9 2 ]+[ 9 2 ] =9

Thus, the value of integral 6 0 | x+3 | dx is 9sq units.


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