Answer
$f(x)$ = $x^{3}$ (1, 1)
ANSWER: $6.75$
Work Step by Step
$f(x)$ = $x^{3}$ (1, 1)
To find the equation of the tangent line, we must first find the derivative:
$f'(x)$ = $3x^{2}$
We can then plug this into the point-slope form of a linear line:
$y - y_{1}$ = $m$($x - x_{1}$)
$y - y_{1}$ = $3x^{2}$($x - x_{1}$)
We then use the point (1, 1) to find the equation of the tangent line:
$y - y_{1}$ = $3x^{2}$($x - x_{1}$)
$y -(1)$ = $3(1)^{2}$($x - (1)$)
$y$ = $3x - 2$
Our integral now looks like this:
$\int$$f(x)-g(x)$ $dx$
$\int$$x^{3}-(3x-2)$ $dx$
To find the upper and lower bounds of the integral, we must find where $f(x)$ and $g(x)$ intersect by setting them equal to each other:
$x^{3}$ = $3x-2$
$x^{3}-3x+2 = 0$
$x = -2, 1$
The integral $\int$$x^{3}-(3x-2)$ $dx$ must now be evaluated from $x = -2$ to $x = 1$, resulting in the answer:
$6.75$
