Question

How do you evaluate the definite integral by the limit definition given \[\int{4dx}\] from \[[-a,a]\]?

Answer 1

Hint: We are given an integral with a definite limit and we have to find the value of this definite integral. If we write the entire expression with the limits, we will have the expression as, \[\int\limits_{-a}^{a}{4dx}\].
Firstly, we will take 4 out and we will have \[4\int\limits_{-a}^{a}{1.dx}\]. Then, as we know that the integration of 1 is x, that is, \[\int{1.dx}=x\]. Applying this to our expression, we will have the expression as, \[4[x]_{-a}^{a}\]. Applying the limits further, we will have the value of the expression.

Complete step by step answer:
According to the question given to us, we have to find the value of the expression which is a definite integral having the specified values.
The expression we have to evaluate is,
\[\int\limits_{-a}^{a}{4dx}\] ----(1)
We will first take the 4 out, so the expression we have will be,
\[\Rightarrow 4\int\limits_{-a}^{a}{1.dx}\]----(2)
We know that, the integration of 1 is x, so we have,
\[\int{1.dx}=x\]
Applying the above in the equation, we will have the expression as,
\[\Rightarrow 4[x]_{-a}^{a}\]
Now, we will apply the limits, we get,
\[\Rightarrow 4[a-(-a)]\]
Solving further, we will have,
\[\Rightarrow 4[2a]\]
On calculating further, we will have the value of the expression as,
\[\Rightarrow 8a\]

Therefore, the value of the given expression is \[8a\].

Note: While carrying out the integration of the given expression, make sure that each of the steps are done correctly and neatly. Also, while applying the limits care should be taken so as not to write it incorrectly.