Find the volume of the solid generated by revolving the region bounded by the graphs of y=x^(2)+3 and y=x+9 about the x-axis. The volume of the solid is ◻ cubic units. (Type an exact answer, using pi as needed.)

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of  y=x^(2)+3 and  y=x+9 about the  x-axis. The volume of the solid is  ◻ cubic units. (Type an exact answer, using  pi as needed.)

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Identify bounds of integration:  Identify the bounds of integration by setting the equations equal to each other: x^2 + 3 = x + 9. Solve for x: x^2 - x - 6 = 0 (x - 3)(x + 2) = 0 x = 3, x = -2Set up integral for volume:  Set up the integral for the volume using the washer method, where the outer radius R(x) is x+9 and the inner radius r(x) is x^2+3: Volume = π ∫[from x=-2 to x=3] [(x+9)^2 - (x^2+3)^2] dxExpand and simplify integrand:  Expand the squares and simplify the integrand: (x+9)^2 = x^2 + 18x + 81 (x^2+3)^2 = x^4 + 6x^2 + 9 Integrand: (x^2 + 18x + 81) - (x^4 + 6x^2 + 9) Simplify: -x^4 - 4x^2 + 18x + 72Integrate function:  Integrate the function from -2 to 3: ∫[-x^4 - 4x^2 + 18x + 72] dx from -2 to 3 = [(-1/5)x^5 - (4/3)x^3 + 9x^2 + 72x] from -2 to 3 = [(-1/5)(3)^5 - (4/3)(3)^3 + 9(3)^2 + 72(3)] - [(-1/5)(-2)^5 - (4/3)(-2)^3 + 9(-2)^2 + 72(-2)] = [-48.6 - 36 + 81 + 216] - [6.4 + 10.67 + 36 - 144] = 212.4 - (-91.07) = 303.47

Find the volume of the solid generated by revolving the region bounded by the graphs of $y=x^{2}+3$ and $y=x+9$ about the $x$ -axis. $\newline$ The volume of the solid is $\square$ cubic units. $\newline$ (Type an exact answer, using $\pi$ as needed.)

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Find the volume of the solid generated by revolving the region bounded by the graphs of y=x2+3 y=x^{2}+3 y=x2+3 and y=x+9 y=x+9 y=x+9 about the x x x-axis.\newlineThe volume of the solid is □ \square □ cubic units.\newline(Type an exact answer, using π \pi π as needed.)

Find the volume of the solid generated by revolving the region bounded by the graphs of $y=x^{2}+3$ and $y=x+9$ about the $x$ -axis. $\newline$ The volume of the solid is $\square$ cubic units. $\newline$ (Type an exact answer, using $\pi$ as needed.)