Question

\[\int_{ - 4}^4 {\left| {x + 2} \right|} dx = \]
A) \[50\]
B) \[24\]
C) \[20\]
D) None of these

Answer 1

Hint: in this question, we have to find the given integral. In order to find this, the properties of modulus and formula of definite integral is used. Break modulus function into two limits one part is for negative limit and other for positive limit.

Formula Used: The definite integral is the area under the curve between two fixed limits.
Let f(x) is a function and suppose integration of function f(x) is F(x) then definite integral of f(x) having upper limit b and lower limit a can be written in mathematical expression as
\[\int_a^b {f(x)} dx = F(b) - F(a)\]
b is upper limit of integral and a is a lower limit of integral.
Property of mod function
If limit is less than zero
\[\left| {f(x)} \right| = - f(x)\]
If limit of function is greater than zero
\[\left| {f(x)} \right| = f(x)\]

Complete step by step solution: Given: Definite integral \[\int_{ - 4}^4 {\left| {x + 2} \right|} dx\]
Here in this integral upper limit is \[4\]and lower limit is \[ - 4\]
\[\int_{ - 4}^4 {\left| {x + 2} \right|} dx = \int_{ - 4}^{ - 2} { - (x + 2)} dx + \int_{ - 2}^4 {(x + 2)} dx\]
\[\int_{ - 4}^{ - 2} { - (x + 2)} dx + \int_{ - 2}^4 {(x + 2)} dx = [\dfrac{{ - {x^2}}}{2} - 2x]_{ - 4}^{ - 2} + [\dfrac{{{x^2}}}{2} + 2x]_{ - 2}^4\]
\[\begin{array}{l}[\dfrac{{ - {x^2}}}{2} - 2x]_{ - 4}^{ - 2} + [\dfrac{{{x^2}}}{2} + 2x]_{ - 2}^4 = [( - 2 + 4) - (8 + 8)] + [(8 + 8) - (2 - 4)]\\\end{array}\]
\[\begin{array}{l}[( - 2 + 4) - (8 + 8)] + [(8 + 8) - (2 - 4)] = 2 + 16 + 2\\\end{array}\]
\[ = 20\]
So required definite integral is
\[20\]

Option ‘C’ is correct

Note: We must remember that mod function always give positive value. So to make modulus positive break the function of modulus into two different limit.
The definite integral is the area under the curve between two fixed limits.
Let f(x) is a function and suppose integration of function f(x) is F(x) then definite integral of f(x) having upper limit b and lower limit a can be written in mathematical expression as
\[\int_a^b {f(x)} dx = F(b) - F(a)\]
Properties of the definite integrals are:
1) Interchanging the upper and lower limit: \[\int_b^a {f(x)} dx = - \int_a^b {f(x)} dx\]
2) \[\int_b^a {f(x)} dx = \int_b^a {f(t)} dt\]
3) \[\int_0^a {f(x)} dx = \int_0^a {f(a - x)} dx\]
4) \[\int_a^b {f(x)} dx = \int_a^c {f(x)} dx + \int_c^b {f(x)} dx\]