graph ( y = x ^ { 2 } + 7 ) using the graphing utility. consider the region ( r ) between ( y = x ^ { 2 } +…

graph ( y = x ^ { 2 } + 7 ) using the graphing utility. consider the region ( r ) between ( y = x ^ { 2 } + 7, x = 0, x = 2 ), and ( y = 0 ). use the shell method or disk method to find the volume ( v ) of the solid formed by rotation ( r ) around the ( y )-axis. (express numbers in exact form. use symbolic notation and fractions where needed.) ( v = )

graph ( y = x ^ { 2 } + 7 ) using the graphing utility. consider the region ( r ) between ( y = x ^ { 2 } + 7, x = 0, x = 2 ), and ( y = 0 ). use the shell method or disk method to find the volume ( v ) of the solid formed by rotation ( r ) around the ( y )-axis. (express numbers in exact form. use symbolic notation and fractions where needed.) ( v = )

Answer

Explanation:

Step1: Recall the Shell Method formula

The Shell Method formula for rotating about the (y -)axis is (V = 2\pi\int_{a}^{b}x\cdot f(x)dx), where (a = 0), (b = 2), and (f(x)=x^{2}+7).

Step2: Set up the integral

Substitute into the formula: (V = 2\pi\int_{0}^{2}x(x^{2}+7)dx=2\pi\int_{0}^{2}(x^{3}+7x)dx).

Step3: Integrate term - by - term

Using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (\int(x^{3}+7x)dx=\frac{x^{4}}{4}+\frac{7x^{2}}{2}+C).

Step4: Evaluate the definite integral

[ \begin{align*} V&=2\pi\left[\left(\frac{x^{4}}{4}+\frac{7x^{2}}{2}\right)\Big|_{0}^{2}\right]\ &=2\pi\left(\frac{2^{4}}{4}+\frac{7\times2^{2}}{2}-0\right)\ &=2\pi\left(4 + 14\right)\ &=2\pi\times18\ &=36\pi \end{align*} ]

Answer:

(V = 36\pi)