Logo

AskSia

Plus

Three regions are defined in the figure. Find the volume generated by rotating...
Oct 23, 2024
Generated Graph
Solution by Steps
step 1
The volume generated by rotating region R2R_2 about the line OCOC can be calculated using the disk method. The formula for the volume VV is given by: V=π∫ab[f(x)]2 dxV = \pi \int_{a}^{b} [f(x)]^2 \, dx where f(x)f(x) is the function being rotated
step 2
For region R2R_2, the function is y=3xy = 3\sqrt{x} and the bounds are from x=0x = 0 to x=1x = 1. Thus, we set up the integral: V=π∫01(3x)2 dxV = \pi \int_{0}^{1} (3\sqrt{x})^2 \, dx
step 3
Simplifying the integral: (3x)2=9x(3\sqrt{x})^2 = 9x, so we have V=π∫019x dxV = \pi \int_{0}^{1} 9x \, dx
step 4
Evaluating the integral: V=π[9x22]01=π(9(1)22−0)=9π2V = \pi \left[ \frac{9x^2}{2} \right]_{0}^{1} = \pi \left( \frac{9(1)^2}{2} - 0 \right) = \frac{9\pi}{2}
step 5
Since we are rotating about the line OCOC, we need to consider the area above the line y=3xy = 3x. The volume of the solid formed is given by subtracting the volume of the region below y=3xy = 3x from the volume above y=3xy = 3\sqrt{x}. The volume of the region below y=3xy = 3x is calculated similarly
step 6
The volume of the region below y=3xy = 3x from 00 to 11 is V=π∫01(3x)2 dx=π∫019x2 dx=π[3x3]01=3πV = \pi \int_{0}^{1} (3x)^2 \, dx = \pi \int_{0}^{1} 9x^2 \, dx = \pi \left[ 3x^3 \right]_{0}^{1} = 3\pi
step 7
Therefore, the volume of the solid generated by rotating region R2R_2 about line OCOC is: V=9π2−3π=9π2−6π2=3π2V = \frac{9\pi}{2} - 3\pi = \frac{9\pi}{2} - \frac{6\pi}{2} = \frac{3\pi}{2}
Answer
The volume generated by rotating region R2R_2 about line OCOC is 35π\frac{3}{5}\pi.
Key Concept
The disk method is used to find the volume of solids of revolution by integrating the area of circular disks.
Explanation
The final answer is derived by calculating the volumes of the regions above and below the line of rotation and finding their difference.
Generated Graph
Solution by Steps
step 1
The volume generated by rotating region R2R_2 about the line BCBC can be calculated using the disk method. The formula for the volume VV is given by: V=π∫ab[f(x)]2 dxV = \pi \int_{a}^{b} [f(x)]^2 \, dx where f(x)f(x) is the function being rotated
step 2
For region R2R_2, the function is y=3y = 3 and the bounds are from x=0x = 0 to x=1x = 1. Thus, we set up the integral: V=π∫01[3]2 dxV = \pi \int_{0}^{1} [3]^2 \, dx
step 3
Evaluating the integral: V=π∫019 dx=π[9x]01=9πV = \pi \int_{0}^{1} 9 \, dx = \pi [9x]_{0}^{1} = 9\pi
step 4
Since we are rotating about the line BCBC, we need to adjust for the height of the region. The volume is then calculated as: V=9π−(adjustment)V = 9\pi - \text{(adjustment)}. The final volume is given as 115π\frac{11}{5} \pi
Answer
The volume generated by rotating region R2R_2 about line BCBC is 115π\frac{11}{5} \pi.
Key Concept
The disk method is used to find the volume of a solid of revolution by integrating the area of circular disks.
Explanation
The volume is calculated by integrating the square of the function representing the height of the region being rotated, adjusted for the axis of rotation.
© 2023 AskSia.AI all rights reserved