Calculus for Team-Based Inquiry Learning: 2022 Edition

(a)

Suppose we estimate the volume of this solid by approximating it with two cylinders of “height” \(\Delta x=3\) with radii \(f(0)=1\) and \(f(3)=4\text{.}\)

What is the volume of the cylinder with radius \(f(0)=1\text{?}\)

1. \(\displaystyle \pi\cdot 0^2\cdot 3\)

2. \(\displaystyle \pi\cdot 1^2\cdot 3\)

3. \(\displaystyle \pi\cdot 3^2\cdot 3\)

4. \(\displaystyle \pi\cdot 4^2\cdot 3\)

5. \(\displaystyle \pi\cdot 6^2\cdot 3\)

6. \(\displaystyle \pi\cdot 7^2\cdot 3\)

(b)

What is the volume of the cylinder with radius \(f(3)=4\text{?}\)

1. \(\displaystyle \pi\cdot 0^2\cdot 3\)

2. \(\displaystyle \pi\cdot 1^2\cdot 3\)

3. \(\displaystyle \pi\cdot 3^2\cdot 3\)

4. \(\displaystyle \pi\cdot 4^2\cdot 3\)

5. \(\displaystyle \pi\cdot 6^2\cdot 3\)

6. \(\displaystyle \pi\cdot 7^2\cdot 3\)

(c)

Suppose we now use a “height” of \(\Delta x=2\text{.}\)

Fill out the following table.

(d)

Which of the following Reimann sums best approximates the volume of our solid?

1. \(\displaystyle \displaystyle \sum \pi x_i^2\Delta x\)

2. \(\displaystyle \displaystyle \sum \pi (x_i+1)^2\Delta x\)

3. \(\displaystyle \displaystyle \sum 2\pi x_i\Delta x\)

4. \(\displaystyle \displaystyle \sum 2\pi (x_i+1)\Delta x\)

(e)

Which of the following integrals computes the volume of our solid?

1. \(\displaystyle \displaystyle \int_0^6 2\pi (x+1)dx\)

2. \(\displaystyle \displaystyle \int_0^6 \pi x^2dx\)

3. \(\displaystyle \displaystyle \int_0^6 2\pi xdx\)

4. \(\displaystyle \displaystyle \int_0^6 \pi (x+1)^2dx\)

(f)

What is the volume of our solid?

(a)

Consider a cross section at height \(y\text{.}\) What is the radius of this cross section?

1. \(\displaystyle r=4\)

2. \(\displaystyle r=2\)

3. \(\displaystyle r=\sqrt{x}\)

4. \(\displaystyle r=x^2\)

5. \(\displaystyle r=\sqrt{y}\)

6. \(\displaystyle r=y^2\)

(b)

Suppose we wanted to estimate this volume with 4 cylinders with \(\Delta y=0.5\text{.}\) Fill out the following table.

(c)

Which of the following Reimann sums best approximates the volume of our solid?

1. \(\displaystyle \displaystyle \sum 2\pi y_i^2\Delta y\)

2. \(\displaystyle \displaystyle \sum \pi (y_i^2)^2\Delta y\)

3. \(\displaystyle \displaystyle \sum 2\pi \sqrt{y_i}\Delta y\)

4. \(\displaystyle \displaystyle \sum \pi (\sqrt{y_i})^2\Delta y\)

(d)

Which of the following integrals computes the volume of our solid?

1. \(\displaystyle \displaystyle \int_0^2 \pi y^4 dy\)

2. \(\displaystyle \displaystyle \int_0^4 \pi y^4 dy\)

3. \(\displaystyle \displaystyle \int_{-4}^4 \pi y^4 dy\)

(e)

What is the volume of our solid?

(a)

What is the shape of the cross section made at \(x=x_i\text{?}\)

1. A circle with radius \(r=x_i\text{.}\)

2. A circle with radius \(r=2x_i+1\text{.}\)

3. A circle with radius \(r=2x_i+1-x_i=x_i+1\text{.}\)

4. An annulus with outside radius \(R=2x_i+1\) and inner radius \(r=x_i\text{.}\)

(b)

Suppose we wanted to estimate this volume with 4 cylinders or annular cylinders with \(\Delta x=1\text{.}\) What is the volume of the (annular) cylinder formed at the cross section \(x_i=1\text{?}\)

1. \(\displaystyle \pi\cdot 1^2\cdot 1\)

2. \(\displaystyle \pi\cdot 3^2\cdot 1\)

3. \(\displaystyle \pi\cdot 2^2\cdot 1\)

4. \(\displaystyle (\pi\cdot 3^2-\pi\cdot 1^2)\cdot 1\)

(c)

We continue to estimate this volume with 4 cylinders or annular cylinders with \(\Delta x=1\text{.}\) Fill out the following table:

(d)

Which of the following Reimann sums best approximates the volume of our solid?

1. \(\displaystyle \displaystyle \sum \pi x_i^2\Delta x\)

2. \(\displaystyle \displaystyle \sum \pi (2x_i+1)^2\Delta x\)

3. \(\displaystyle \displaystyle \sum \pi (2x_i+1-x_i)^2\Delta x\)

4. \(\displaystyle \displaystyle \sum \pi \left((2x_i+1)^2-x_i^2 \right)\Delta x\)

(e)

Which of the following integrals computes the volume of our solid?

1. \(\displaystyle \displaystyle \int_0^4 \pi x^2 dx\)

2. \(\displaystyle \displaystyle \int_0^4 \pi (2x+1)^2 dx\)

3. \(\displaystyle \displaystyle \int_0^4 \pi (x+1)^2 dx\)

4. \(\displaystyle \displaystyle \int_0^4 \pi ((2x+1)^2-x^2) dx\)

(f)

What is the volume of our solid?

(g)

How else could we have computed the volume?

1. Find the volume of the region bounded by \(f(x)=2x+1\) on \([0,4]\) rotated about the \(x\)-axis, and subtract the volume of the region bounded by \(g(x)=x\) on \([0,4]\) rotated about the \(x\)-axis.

2. Find the volume of the region bounded by \(f(x)=2x+1-x=x+1\) on \([0,4]\) rotated about the \(x\)-axis.

(a)

Suppose we wanted to estimate this volume with 3 concentric annular cylinders with heights \(h_i=f(r_i)=\frac{4}{2^{r_i}}\text{,}\) where \(r_i\) are the inner radii and \(R_i\) are the outer radii. Over which values do the radii \(r_i\) and \(R_i\) range?

1. \(\displaystyle [-3,3]\)

2. \(\displaystyle [0,3]\)

3. \(\displaystyle [-4,4]\)

4. \(\displaystyle [0,4]\)

(b)

Recall that the volume of an annular cylinder of height \(h\text{,}\) inner radius \(r\) and outer radius \(R\) is \(V=\pi(R^2-r^2)h\text{.}\)

Fill out the following table.

(c)

Consider an arbitrary annular cylinder with inner radius \(r_i\text{,}\) outer radius \(r_i+\Delta r\text{,}\) and height \(\displaystyle h_i=\frac{4}{2^{r_i}}\text{.}\)

Which of the following represents the volume of this annular cylinder?

1. \(\displaystyle \pi r_i^2\cdot \frac{4}{2^{r_i}}\)

2. \(\displaystyle \pi ((r_i+\Delta r)^2-r_i^2)\cdot \frac{4}{2^{r_i}}\)

3. \(\displaystyle \pi ((r_i+\Delta r)-r_i)^2\cdot \frac{4}{2^{r_i}}\)

4. \(\displaystyle \pi (r_i+\Delta r)^2\cdot \frac{4}{2^{r_i+\Delta r}}\)

(d)

Recall that \(\pi ((r_i+\Delta r)^2-r_i^2)=\pi(2(\Delta r) r_i+(\Delta r)^2)=\pi(2 r_i+(\Delta r))\Delta r\text{.}\) Which of the following Riemann sums best estimates the volume of our solid?

1. \(\displaystyle \displaystyle \sum \pi(2 r_i+(\Delta r))\frac{4}{2^{r_i}}\Delta r\)

2. \(\displaystyle \displaystyle \sum \pi r_i^2\frac{4}{2^{r_i}}\Delta r\)

3. \(\displaystyle \displaystyle \sum \pi (r_i+\Delta r)^2\frac{4}{2^{r_i}}\Delta r\)

(e)

Which of the following integrals computes the volume of our solid?

1. \(\displaystyle \displaystyle \int_{-3}^3 2\pi r \frac{4}{2^r} dr\)

2. \(\displaystyle \displaystyle \int_{0}^3 2\pi r \frac{4}{2^r} dr\)

3. \(\displaystyle \displaystyle \int_{0}^3 2\pi r \frac{4}{2^r} dr\)

(f)

What is the volume of our solid?

(a)

Find the volume of the solid generated by rotating this region about the \(x\)-axis.

(b)

Find the volume of the solid generated by rotating this region about the \(y\)-axis.