Question

Sketch the graph of and evaluate

Answer 1

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.

x	y
−6	3
−5	2
−4	1
−3	0
−2	1
−1	2
0	3

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 ) −6≤x≤−3 y=( x+3 ) −3≤x≤0

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 −9−18+18 ]+[ − 9 2 +9 ]

Further, solve the above equation.

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

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