Question

3 region r is bounded by the lines y = 4, x = 1, x = 6 and the x-axis. region r is rotated about the x-axis.

Explanation:

Step1: Identify the shape of Region R

Region R is bounded by \( y = 4 \), \( x = 1 \), \( x = 6 \), and the \( x \)-axis. So it's a rectangle with height \( y = 4 \) and width \( 6 - 1 = 5 \) along the \( x \)-axis. When rotated about the \( x \)-axis, it forms a cylinder. The formula for the volume of a solid of revolution using the disk method is \( V=\pi\int_{a}^{b}[f(x)]^{2}dx \), where \( f(x) \) is the radius of the disk at position \( x \). Here, \( f(x)=4 \) (since the distance from the \( x \)-axis to \( y = 4 \) is the radius), \( a = 1 \), and \( b = 6 \).

Step2: Set up the integral

Substitute \( f(x)=4 \), \( a = 1 \), and \( b = 6 \) into the disk method formula:
\( V=\pi\int_{1}^{6}(4)^{2}dx \)
Simplify \( (4)^{2}=16 \), so the integral becomes \( V=\pi\int_{1}^{6}16dx \).

Step3: Evaluate the integral

The integral of a constant \( 16 \) with respect to \( x \) is \( 16x \). Evaluate from \( 1 \) to \( 6 \):
\( \int_{1}^{6}16dx=16x\big|_{1}^{6}=16(6)-16(1)=96 - 16 = 80 \).

Step4: Find the volume

Multiply by \( \pi \): \( V=\pi\times80 = 80\pi \).

Answer:

The volume of the solid formed by rotating Region R about the \( x \)-axis is \( \boldsymbol{80\pi} \) (or approximately \( 251.33 \) if a decimal approximation is needed).