The volume of the solid generated by revolving $$ riangle A B C $$ such that $$ A(-2,0), B(1,5), C(4,0) $$ a complete revolution about $$ \boldsymbol{x} $$-axis $$ =\ldots $$ cubic unit. $$ \begin{array}{llll} ext { (a) } 25 \pi & ext { (b) } 50 \pi & ext { (C) } 75 \pi & ext { (d) } 90 \pi\end{array} $$

Question

The volume of the solid generated by revolving $$ 	riangle A B C $$ such that $$ A(-2,0), B(1,5), C(4,0) $$ a complete revolution about $$ \boldsymbol{x} $$-axis $$ =\ldots $$ cubic unit. $$ \begin{array}{llll}	ext { (a) } 25 \pi & 	ext { (b) } 50 \pi & 	ext { (C) } 75 \pi & 	ext { (d) } 90 \pi\end{array} $$

UpStudy AI · Accepted Answer

To find the volume of the solid generated by revolving triangle $$ \triangle ABC $$ about the $$ x $$-axis, we can use the method of disks (washers).

**Given:**
- Vertices of the triangle: $$ A(-2, 0) $$, $$ B(1, 5) $$, and $$ C(4, 0) $$.

**Steps:**

1. **Divide the Triangle:**
   - The triangle can be divided into two segments:
     - From $$ x = -2 $$ to $$ x = 1 $$ (segment AB).
     - From $$ x = 1 $$ to $$ x = 4 $$ (segment BC).

2. **Find the Equations of the Lines:**
   - **Segment AB:**
     - Slope $$ m_1 = \frac{5 - 0}{1 - (-2)} = \frac{5}{3} $$.
     - Equation: $$ y = \frac{5}{3}(x + 2) $$.
   - **Segment BC:**
     - Slope $$ m_2 = \frac{0 - 5}{4 - 1} = -\frac{5}{3} $$.
     - Equation: $$ y = -\frac{5}{3}x + \frac{20}{3} $$.

3. **Set Up the Integral for Volume:**
   - The volume $$ V $$ is given by:
     $$
     V = \pi \int_{-2}^{1} \left(\frac{5}{3}(x + 2)\right)^2 dx + \pi \int_{1}^{4} \left(-\frac{5}{3}x + \frac{20}{3}\right)^2 dx
     $$

4. **Compute the Integrals:**
   - **First Integral:**
     $$
     \pi \left(\frac{25}{9}\right) \int_{-2}^{1} (x + 2)^2 dx = \pi \left(\frac{25}{9}\right) \times 9 = 25\pi
     $$
   - **Second Integral:**
     $$
     \pi \left(\frac{25}{9}\right) \int_{1}^{4} (x - 4)^2 dx = \pi \left(\frac{25}{9}\right) \times 9 = 25\pi
     $$
   - **Total Volume:**
     $$
     V = 25\pi + 25\pi = 50\pi
     $$

**Answer:**

**(b) $$50 \pi$$**
The volume of the solid is $$50\pi$$ cubic units.

The volume of the solid generated by revolving such that
a complete revolution about -axis cubic unit.

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The volume of the solid generated by revolving such that a complete revolution about -axis cubic unit.