Use both the washer method and the shell method to find the volume of the solid that is generated when the region
in the first quadrant bounded by , and is revolved about the line .
Set up the integral that gives the volume of the solid as a single integral if possible using the disk/washer method.
Select the correct choice below and fill in any answer boxes within your choice.
(Type exact answers.)

Ask by Bolton Cook. in the United States

Mar 29,2025

Question

Use both the washer method and the shell method to find the volume of the solid that is generated when the region
in the first quadrant bounded by , and is revolved about the line .
Set up the integral that gives the volume of the solid as a single integral if possible using the disk/washer method.
Select the correct choice below and fill in any answer boxes within your choice.
(Type exact answers.)

Ask by Bolton Cook. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

$$
\textbf{Washer Method:}
$$

We take vertical slices at a fixed $$ x $$ in the interval $$ 0 \le x \le 5 $$ (since $$ y=x^2 $$ meets $$ y=25 $$ when $$ x=5 $$). For each $$ x $$, the region has $$ y $$ running from $$ y=x^2 $$ (inner curve) up to $$ y=25 $$ (outer boundary). When this slice is revolved about the horizontal line $$ y=-4 $$, it produces a washer with

- Outer radius: the distance from the axis $$ y=-4 $$ to $$ y=25 $$  
  $$
  R = 25 - (-4)=29,
  $$
- Inner radius: the distance from $$ y=-4 $$ to $$ y=x^2 $$  
  $$
  r = x^2 - (-4)=x^2+4.
  $$

Thus the volume is given by

$$
V = \pi \int_{0}^{5} \Big[(29)^2 - (x^2+4)^2\Big]\,dx.
$$

---

$$
\textbf{Shell Method:}
$$

We now take horizontal slices at a fixed $$ y $$. In the region the $$ y $$-values go from the lowest point $$ y=0 $$ (when $$ x=0 $$ on the curve $$ y=x^2 $$) up to $$ y=25 $$. For a fixed $$ y $$, $$ x $$ runs from $$ x=0 $$ (left boundary) to $$ x=\sqrt{y} $$ (obtained from $$ y=x^2 $$). When this slice (of thickness $$ dy $$) is rotated about the line $$ y=-4 $$, it forms a cylindrical shell. Its characteristics are:

- Radius: the distance from the slice at height $$ y $$ to the axis $$ y=-4 $$ is
  $$
  y - (-4)=y+4.
  $$
- Height: given by the horizontal distance 
  $$
  h = \sqrt{y} - 0 = \sqrt{y}.
  $$

Thus the volume using the shell method is

$$
V = 2\pi \int_{0}^{25} (y+4)\sqrt{y}\,dy.
$$

---

Both integrals represent the volume of the solid formed by revolving the given region about $$ y=-4 $$.
To find the volume of the solid generated by revolving the region bounded by $$ y = x^2 $$, $$ y = 25 $$, and $$ x = 0 $$ about the line $$ y = -4 $$, we can use both the washer method and the shell method. **Washer Method:** - **Outer Radius ($$ R $$)**: Distance from the axis $$ y = -4 $$ to $$ y = 25 $$ is $$ 29 $$. - **Inner Radius ($$ r $$)**: Distance from $$ y = -4 $$ to $$ y = x^2 $$ is $$ x^2 + 4 $$. - **Volume Integral**: $$ V = \pi \int_{0}^{5} \Big[(29)^2 - (x^2 + 4)^2\Big]\,dx $$ **Shell Method:** - **Radius ($$ r $$)**: Distance from the slice at height $$ y $$ to the axis $$ y = -4 $$ is $$ y + 4 $$. - **Height ($$ h $$)**: $$ \sqrt{y} $$. - **Volume Integral**: $$ V = 2\pi \int_{0}^{25} (y + 4)\sqrt{y}\,dy $$ Both integrals calculate the volume of the solid formed by the revolution.

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