Find the volume of the solid obtained by rotating the
region bounded by the given curves about the specified
axis.
x + y = 5, x = 6 − (y − 1)2
;
about the x-axis.
Volume =


To find the volume of the solid obtained by rotating the region bounded by the curves \(x + y = 5\) and \(x = 6 - (y - 1)^2\) about the x-axis, we can use the method of cylindrical shells. Here is the step-by-step process:

### Step 1: Set up the problem

We have the following curves:
1. \( x + y = 5 \) or rewritten as \( y = 5 - x \),
2. \( x = 6 - (y - 1)^2 \) or rewritten as \( y = 1 + \sqrt{6 - x} \) and \( y = 1 - \sqrt{6 - x} \) (since the equation is quadratic).

### Step 2: Determine the limits of integration
We need to find the points where the two curves intersect. Set \( y = 5 - x \) equal to \( y = 1 + \sqrt{6 - x} \) and solve for \(x\).

\[
5 - x = 1 + \sqrt{6 - x}
\]
\[
4 - x = \sqrt{6 - x}
\]
Square both sides:
\[
(4 - x)^2 = 6 - x
\]
\[
16 - 8x + x^2 = 6 - x
\]
\[
x^2 - 7x + 10 = 0
\]
Solve this quadratic equation using the quadratic formula:
\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(10)}}{2(1)} = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm \sqrt{9}}{2} = \frac{7 \pm 3}{2}
\]
Thus, \(x = 5\) and \(x = 2\).

### Step 3: Set up the integral

The volume of the solid obtained by rotating the region about the x-axis is given by the formula:

\[
V = \pi \int_{x_1}^{x_2} \left( f(y)^2 - g(y)^2 \right) \, dy
\]

I will compute the final

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x + y = 5, x = 6 − (y − 1)^2 ; about the x-axis.

Learn how to find the volume of a solid formed by rotating the curves x + y = 5 and x = 6 - (y - 1)^2 around the x-axis. This detailed solution covers key calculus concepts, including the cylindrical shell method and quadratic equations. Suitable for advanced high school and college students.

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