Question

Identify each expression and value that represents the area under the curve y=x^2+4 on the interval [-3,2]

Answer 1

The area under a graph is given by the integral of that function, evaluated in the interval of interest:

`\displaystyle \int_(-3)^2 x^2+4\;dx = \left[(x^3)/(3)+4x\right]_(-3)^2 = \left[(2^3)/(3)+4\cdot 2\right]-\left[((-3)^3)/(3)+4\cdot(-3)\right] = \left[(8)/(3)+8\right]-\left[-9-12\right] = (32)/(3)+21 = (95)/(3)`

Answer 2

`((95)/(3))`

Explanation:

Equation that represents curve is

y = x² + 4

and we have to find the area

under the curve in the interval of [-3, 2] will be

area =
`\int\limits^2_ {-3} \,x^(2)+4 dx`

=
`[(x^(3) )/(3)+4x ]^(2) _(-3)`

=
`(1)/(3)[x^(3)]^2_(-3) +4[x]^2_-3`

=
`(1)/(3)[(2)^3(-3)^3]+4[(2)-(-3)]`

=
`(1)/(3)[8+27]+4[2+3]`

=
`(1)/(3)(35)+20`

=
`((95)/(3))`