Math  /  Calculus

QuestionThree regions are defined in the figure.
Find the volume generated by rotating the given region about the specified line. R1 about OA\mathscr{R}_{1} \text { about } O A \square

Studdy Solution

STEP 1

1. The region R1 \mathscr{R}_1 is a right triangle with vertices at O(0,0) O(0,0) , A(1,0) A(1,0) , and B(1,3) B(1,3) .
2. The line OA OA is the x-axis.
3. We are using the method of disks or washers to find the volume of the solid of revolution.

STEP 2

1. Set up the integral for the volume of the solid of revolution.
2. Evaluate the integral to find the volume.

STEP 3

Identify the function that represents the boundary of the region being rotated. The line y=3x y = 3x is the boundary of R1 \mathscr{R}_1 .

STEP 4

Determine the limits of integration. The region is bounded between x=0 x = 0 and x=1 x = 1 .

STEP 5

Set up the integral using the disk method. The volume V V is given by:
V=π01(3x)2dx V = \pi \int_{0}^{1} (3x)^2 \, dx

STEP 6

Evaluate the integral:
V=π019x2dx V = \pi \int_{0}^{1} 9x^2 \, dx
V=9π01x2dx V = 9\pi \int_{0}^{1} x^2 \, dx
V=9π[x33]01 V = 9\pi \left[ \frac{x^3}{3} \right]_{0}^{1}
V=9π(133033) V = 9\pi \left( \frac{1^3}{3} - \frac{0^3}{3} \right)
V=9π(13) V = 9\pi \left( \frac{1}{3} \right)
V=3π V = 3\pi
The volume of the solid generated by rotating the region R1 \mathscr{R}_1 about the line OA OA is:
3π \boxed{3\pi}

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