Answer:
The area of the region is 25,351 [tex]units^2[/tex].
Step-by-step explanation:
The Fundamental Theorem of Calculus: if [tex]f[/tex] is a continuous function on [tex][a,b][/tex], then
[tex]\int_{a}^{b} f(x)dx = F(b) - F(a) = F(x) | {_a^b}[/tex]
where [tex]F[/tex] is an antiderivative of [tex]f[/tex].
A function [tex]F[/tex] is an antiderivative of the function [tex]f[/tex] if
[tex]F^{'}(x)=f(x)[/tex]
The theorem relates differential and integral calculus, and tells us how we can find the area under a curve using antidifferentiation.
To find the area of the region between the graph of the function [tex]x^5 + 8x^4 + 2x^2 + 5x + 15[/tex] and the x-axis on the interval [-6, 6] you must:
Apply the Fundamental Theorem of Calculus
[tex]\int _{-6}^6(x^5+8x^4+2x^2+5x+15)dx[/tex]
[tex]\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\\\\int _{-6}^6x^5dx+\int _{-6}^68x^4dx+\int _{-6}^62x^2dx+\int _{-6}^65xdx+\int _{-6}^615dx[/tex]
[tex]\int _{-6}^6x^5dx=0\\\\\int _{-6}^68x^4dx=\frac{124416}{5}\\\\\int _{-6}^62x^2dx=288\\\\\int _{-6}^65xdx=0\\\\\int _{-6}^615dx=180\\\\0+\frac{124416}{5}+288+0+18\\\\\frac{126756}{5}\approx 25351.2[/tex]
