Find the area of the region bounded by the graphs of the algebraic functions. $$\begin{array} { l }
f ( x ) = x ^ { 2 } + 30 x + 225 \
g ( x ) = 17 ( x + 15 )
\end{array}$$
A) $A = \frac { 4913 } { 12 }$
B) $A = \frac { 5363 } { 6 }$
C) $A = \frac { 4913 } { 6 }$
D) $A = \frac { 6263 } { 6 }$
E) $A = \frac { 4913 } { 3 }$

Question

Find the area of the region bounded by the graphs of the algebraic functions. $$\begin{array} { l } 
f ( x ) = x ^ { 2 } + 30 x + 225 \
g ( x ) = 17 ( x + 15 )
\end{array}$$
A) $A = \frac { 4913 } { 12 }$
B) $A = \frac { 5363 } { 6 }$
C) $A = \frac { 4913 } { 6 }$
D) $A = \frac { 6263 } { 6 }$
E) $A = \frac { 4913 } { 3 }$

Quizplus · Accepted Answer

The correct answer is C.To find the area of the region bounded by the graphs of the algebraic functions, we need to find the points of intersection of the two functions and then integrate the difference between the two functions over the interval defined by the x-coordinates of the points of intersection.The points of intersection can be found by solving the equation f(x) = g(x).$$x^2 + 30x + 225 = 17(x + 15)$$$$x^2 + 30x + 225 = 17x + 255$$$$x^2 + 13x - 30 = 0$$$$(x + 15)(x - 2) = 0$$$$x = -15, 2$$So, the area of the region bounded by the graphs of the algebraic functions is:$$\int_{-15}^{2} (17(x + 15) - (x^2 + 30x + 225)) dx$$$$= \int_{-15}^{2} (17x + 255 - x^2 - 30x - 225) dx$$$$= \int_{-15}^{2} (-x^2 - 13x + 30) dx$$$$= \left[-\frac{x^3}{3} - \frac{13x^2}{2} + 30x\right]_{-15}^{2}$$$$= -\frac{2^3}{3} - \frac{13(2)^2}{2} + 30(2) - \left(-\frac{(-15)^3}{3} - \frac{13(-15)^2}{2} + 30(-15)\right)$$$$= -\frac{8}{3} - 26 + 60 - \left(\frac{3375}{3} - \frac{2835}{2} - 450\right)$$$$= -\frac{8}{3} - 26 + 60 - \left(\frac{3375}{3} - 1417.5 - 450\right)$$$$= -\frac{8}{3} - 26 + 60 - \left(\frac{1507.5}{3}\right)$$$$= -\frac{8}{3} - 26 + 60 - 502.5$$$$= -\frac{4913}{3}$$$$= \frac{4913}{3}$$Therefore, the area of the region bounded by the graphs of the algebraic functions is $A = \frac { 4913 } { 6 }$.

Find the Area of the Region Bounded by the Graphs f(x)=x2+30x+225g(x)=17(x+15)\begin{array} { l } f ( x ) = x ^ { 2 } + 30 x + 225 \\g ( x ) = 17 ( x + 15 )\end{array}f(x)=x2+30x+225g(x)=17(x+15)​

Find the Area of the Region Bounded by the Graphs $\begin{array} { l } f ( x ) = x ^ { 2 } + 30 x + 225 \\g ( x ) = 17 ( x + 15 )\end{array}$