Question

find the volume of the solid formed by rotating the region enclosed by

x = 0, x = 1, y = 0, y = 7 + x^5

about the x-axis.

Explanation:

Step1: Identify the method

To find the volume of a solid formed by rotating a region about the x - axis, we can use the disk method. The formula for the volume \( V \) using the disk method is \( V=\pi\int_{a}^{b}[f(x)]^{2}dx \), where \( [a,b] \) is the interval of integration and \( y = f(x) \) is the function that bounds the region above the x - axis. In this case, the region is enclosed by \( x = 0 \), \( x=1 \), \( y = 0 \) and \( y=7 + x^{5} \). So \( a = 0 \), \( b = 1 \) and \( f(x)=7 + x^{5} \).

Step2: Set up the integral

Substitute \( f(x)=7 + x^{5} \), \( a = 0 \) and \( b = 1 \) into the disk method formula:
\( V=\pi\int_{0}^{1}(7 + x^{5})^{2}dx \)

Step3: Expand the integrand

Expand \( (7+x^{5})^{2} \) using the formula \( (a + b)^{2}=a^{2}+2ab + b^{2} \), where \( a = 7 \) and \( b=x^{5} \).
\( (7 + x^{5})^{2}=7^{2}+2\times7\times x^{5}+(x^{5})^{2}=49 + 14x^{5}+x^{10} \)

Step4: Integrate term - by - term

We know that \( \int x^{n}dx=\frac{x^{n + 1}}{n+1}+C \) (for \( n
eq - 1 \)).
\( \int_{0}^{1}(49 + 14x^{5}+x^{10})dx=\int_{0}^{1}49dx+\int_{0}^{1}14x^{5}dx+\int_{0}^{1}x^{10}dx \)

• For \( \int_{0}^{1}49dx \): \( \int_{0}^{1}49dx=49x\big|_{0}^{1}=49(1 - 0)=49 \)
• For \( \int_{0}^{1}14x^{5}dx \): \( 14\int_{0}^{1}x^{5}dx=14\times\frac{x^{6}}{6}\big|_{0}^{1}=14\times(\frac{1^{6}}{6}-\frac{0^{6}}{6})=\frac{14}{6}=\frac{7}{3} \)
• For \( \int_{0}^{1}x^{10}dx \): \( \int_{0}^{1}x^{10}dx=\frac{x^{11}}{11}\big|_{0}^{1}=\frac{1^{11}}{11}-\frac{0^{11}}{11}=\frac{1}{11} \)

Step5: Sum the integrals and multiply by \( \pi \)

First, sum the three integrals:
\( 49+\frac{7}{3}+\frac{1}{11}=\frac{49\times33+7\times11 + 3}{33}=\frac{1617+77 + 3}{33}=\frac{1697}{33} \)
Then multiply by \( \pi \):
\( V=\pi\times\frac{1697}{33}=\frac{1697\pi}{33}\approx\frac{1697\times3.1416}{33}\approx\frac{5330.3}{33}\approx161.52 \) (approximate value)

Answer:

The volume of the solid is \( \frac{1697\pi}{33} \) (or approximately \( 161.52 \))