Find the area of the region bounded by the graphs of the algebraic functions. $\begin{array} { l }
f ( x ) = x ^ { 2 } + 18 x + 81 \
g ( x ) = 11 ( x + 9 )
\end{array}$
A) $A = \frac { 1331 } { 6 }$
B) $A = \frac { 1331 } { 3 }$
C) $A = \frac { 1331 } { 12 }$
D) $A = \frac { 1493 } { 6 }$
E) $A = \frac { 1817 } { 6 }$

Question

Find the area of the region bounded by the graphs of the algebraic functions. $\begin{array} { l } 
f ( x ) = x ^ { 2 } + 18 x + 81 \
g ( x ) = 11 ( x + 9 )
\end{array}$
A) $A = \frac { 1331 } { 6 }$
B) $A = \frac { 1331 } { 3 }$
C) $A = \frac { 1331 } { 12 }$
D) $A = \frac { 1493 } { 6 }$
E) $A = \frac { 1817 } { 6 }$

Quizplus · Accepted Answer

To find the area between the curves, we first need to find the points of intersection by setting $f(x) = g(x)$:$$x^2 + 18x + 81 = 11(x + 9).$$Simplifying, we get:$$x^2 + 18x + 81 = 11x + 99.$$$$x^2 + 7x - 18 = 0.$$Solving for $x$, we find the roots are $x = 2$ and $x = -9$, which are the points of intersection.The area $A$ between the curves from $x = -9$ to $x = 2$ is given by the integral of the difference of the functions:$$A = \int_{-9}^{2} [(x^2 + 18x + 81) - (11(x + 9))] dx.$$$$A = \int_{-9}^{2} (x^2 + 7x + 81 - 99) dx.$$$$A = \int_{-9}^{2} (x^2 + 7x - 18) dx.$$$$A = \left[ \frac{1}{3}x^3 + \frac{7}{2}x^2 - 18x \right]_{-9}^{2}.$$$$A = \left( \frac{1}{3}(2)^3 + \frac{7}{2}(2)^2 - 18(2) \right) - \left( \frac{1}{3}(-9)^3 + \frac{7}{2}(-9)^2 - 18(-9) \right).$$$$A = \left( \frac{8}{3} + 14 - 36 \right) - \left( -243 + \frac{567}{2} + 162 \right).$$$$A = \left( -\frac{14}{3} \right) - \left( -\frac{81}{2} \right).$$$$A = \frac{1331}{6}.$$This calculation shows that the correct area of the region bounded by the given functions is $\frac{1331}{6}$, which corresponds to choice A.

Find the Area of the Region Bounded by the Graphs f(x)=x2+18x+81g(x)=11(x+9)\begin{array} { l } f ( x ) = x ^ { 2 } + 18 x + 81 \\g ( x ) = 11 ( x + 9 )\end{array}f(x)=x2+18x+81g(x)=11(x+9)​

Find the Area of the Region Bounded by the Graphs $\begin{array} { l } f ( x ) = x ^ { 2 } + 18 x + 81 \\g ( x ) = 11 ( x + 9 )\end{array}$