Estimate the area under the graph off(x) = 9 + 4x^2 from x = −1 to x = 2

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You need to evaluate the area under the graph of f(x), from `x=-1`  to `x=2` , hence, you should use the definite integral such that:

`int_(-1)^2(9 + 4x^2) dx`

Using the linearity of integrals yields:

`int_(-1)^2 (9 + 4x^2) dx = int_(-1)^2 (9) dx+ int_(-1)^2 (4x^2) dx`

`int_(-1)^2 (9 + 4x^2) dx = 9x|_(-1)^2 + 4x^3/3|_(-1)^2`

You need to use the fundamental theorem of calculus such that:

`int_(-1)^2 (9 + 4x^2) dx = 9(2 - (-1)) + 4(2^3/3 - (-1)^3/3)`

`int_(-1)^2 (9 + 4x^2) dx = 27 + 12`

`int_(-1)^2 (9 + 4x^2) dx = 39 `

Hence, evaluating the area under the graph of function f(x), from `x=-1 to x=2` , yields `int_(-1)^2 (9 + 4x^2) dx = 39.`

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