The region bounded by a parabola x = 3 + 2y - y^2 and a straight line x = 3y - 3 which lies in the first quadrant is revolved about the Y-axis to generate a solid. Find the volume... The region bounded by a parabola x = 3 + 2y - y^2 and a straight line x = 3y - 3 which lies in the first quadrant is revolved about the Y-axis to generate a solid. Find the volume of the solid of revolution. Read more

Steps to Solve

1. Find the points of intersection To determine the volume of the solid generated, we need to first find where the parabola $x = 3 + 2y - y^2$ intersects the line $x = 3y - 3$.

Set the two equations equal: $$ 3 + 2y - y^2 = 3y - 3 $$

Rearranging this gives: $$ y^2 - y - 6 = 0 $$

Factoring gives: $$ (y - 3)(y + 2) = 0 $$

Thus, the solutions are $y = 3$ and $y = -2$. Since we are in the first quadrant, we take $y = 3$.

2. Find the volume using the washer method For revolving around the Y-axis, we use the washer method. The volume $V$ is given by: $$ V = \pi \int_{a}^{b} (R(y)^2 - r(y)^2) , dy $$

Here, $R(y)$ is the outer radius (from the parabola) and $r(y)$ is the inner radius (from the line).

• For the parabola, solve for $x$: $$ R(y) = 3 + 2y - y^2 $$

• For the line, solve for $x$: $$ r(y) = 3y - 3 $$

3. Set up the integral The bounds for $y$ are from $0$ to $3$ (since the region lies in the first quadrant): $$ V = \pi \int_{0}^{3} \left((3 + 2y - y^2)^2 - (3y - 3)^2\right) , dy $$

4. Evaluate the integral Expand and simplify the integrand: $$ (3 + 2y - y^2)^2 = 9 + 12y + 4y^2 + y^4 - 6y^2 $$ $$ (3y - 3)^2 = 9y^2 - 18y + 9 $$

Now substitute these into the integral: $$ V = \pi \int_{0}^{3} \left((9 + 12y + 4y^2 - 6y^2 + y^4) - (9y^2 - 18y + 9)\right) , dy $$

This simplifies to: $$ V = \pi \int_{0}^{3} \left(y^4 - 11y^2 + 30y\right) , dy $$

5. Calculate the definite integral Integrate term by term: $$ \int y^4 dy = \frac{y^5}{5} $$ $$ \int 11y^2 dy = \frac{11y^3}{3} $$ $$ \int 30y dy = 15y^2 $$

Evaluating from $0$ to $3$, we get: $$ V = \pi \left[\left(\frac{3^5}{5} - \frac{11 \cdot 3^3}{3} + 15 \cdot 3^2\right) - 0\right] $$ Calulating this results in: $$ V = \pi \left[\frac{243}{5} - 99 + 135\right] = \pi \left[\frac{243}{5} + 36\right] = \pi \left[\frac{243 + 180}{5}\right] = \frac{423\pi}{5} $$