Question

Find the area of the region enclosed by​ f(x) and the​ x-axis for the given function over the specified interval.

Answer 1

Explanation:

Complete Question

The complete question is shown on the first uploaded image

Answer:

The area is
`A =8 sq\cdot unit`

Explanation:

From the question we are told that

The first equation is
`f(x) = x^2 + x \ \ \ x< 1`

`on[ -2 , 3 ]`

The second equation is
`f(x) = 2 x \ \ \ x \ge 1`

This means that the limit of the area under the enclosed region is limited between -2 to 1 on the x- axis for first equation and 1 to 3 for second equation

Now the area under the region is evaluated as

`A = \int\limits^1_(-2){x^2 + x } \, dx + \int\limits^3_(1){2x } \, dx`

`A ={ (x^3)/(3) + (x^2)/(2) + c } | \left \ 1 } \atop {-2}} \right. + {(2x^2)/(2) }| \left \ 3} \atop {1}} \right.`

`A =9 + c - 1 -c`

`A =8 sq\cdot unit`