Question

Find the area of the region enclosed by f(x) and the x-axis for the given function over the specified interval. x2 + 2x + 2 x2 The area is 54 (Type an integer or a simplified fraction.)

Answer 1

To find this area, it is necessary to solve an integral, actually the sum of 2 integrals

`\int (x^2+2x+2)dx+\int (3x+4)dx`

The first one must be evaluated from -3 to 2 and the second one from 2 to 3

`\begin{gathered} \int (x^2+2x+2)dx+\int (3x+4)dx \\ ((x^3)/(3)+x^2+2x)+((3x^2)/(2)+4x) \\ \end{gathered}`

Evaluate the first integral

`\begin{gathered} (x^3)/(3)+x^2+2x\text{ (From -3 to 2)} \\ ((2^3)/(3)+2^2+2\cdot2)-(((-3)^3)/(3)+(-3)^2+(2\cdot-3)) \\ (8)/(3)+4+4-(-(27)/(3)+9-6) \\ (35)/(3)+5=(50)/(3) \end{gathered}`

Evaluate the second integral

`\begin{gathered} (3x^2)/(2)+4x\text{ (From 2 to 3)} \\ ((3\cdot(3^2))/(2)+4\cdot3)-((3\cdot(2^2))/(2)+4\cdot2) \\ ((27)/(2)+12)-((12)/(2)+8) \\ (15)/(2)+4=(23)/(2) \end{gathered}`

Now, solve the sum

`\begin{gathered} (50)/(3)+(23)/(2) \\ (100+69)/(6)=(169)/(6) \end{gathered}`

The area is 169/6