Find the area of the region bounded by the given curves.
$$y = x ^ { 2 } - 6 x , y = 5 x + 12$$
A) 2197
B) $\frac { 25 } { 3 }$
C) $\frac { 2197 } { 3 }$
D) $\frac { 2197 } { 6 }$
E) 6

Question

Quizplus · Accepted Answer

To find the area between the curves, first find the points of intersection by setting $x^2 - 6x = 5x + 12$. Solving gives $x^2 - 11x - 12 = 0$, which factors to $(x - 12)(x + 1) = 0$, so $x = 12$ and $x = -1$. The area is the integral from $-1$ to $12$ of $(5x + 12) - (x^2 - 6x)$ dx, which simplifies to $\int_{-1}^{12} (-x^2 + 11x + 12) \, dx$. Evaluating this integral gives $\frac{2197}{6}$.

Find the Area of the Region Bounded by the Given y=x2−6x,y=5x+12y = x ^ { 2 } - 6 x , y = 5 x + 12y=x2−6x,y=5x+12

Find the Area of the Region Bounded by the Given $y = x ^ { 2 } - 6 x , y = 5 x + 12$