8.1.4Which way are we going?

Consider the solid generated by rotating the region bounded by the function $y = x^2$ and the $y = 10$ about the $y$-axis. Examine the typical slice as shown in the diagram at right.

1. What type of slice is formed? What is the thickness of each slice?

2. What is the radius of the typical slice?

3. Write and evaluate an integral to calculate the volume of the solid for $0 ≤ y ≤ 10$.

4. Xavier prefers to use functions in terms of $x$. Using the inverse, he obtains the graph at right. Using Xavier’s diagram, set up the integral and show that his method will give the same volume. 

Does it matter about which axis we rotate a region? Consider the region bounded by $y = x^2$ and $y = 2x$.

1. Imagine the different solids formed when the shaded region is rotated about the $x$-axis and the $y$-axis. Will the volumes be equal? If not, which do you predict will be greater and why?

2. Sketch the solid and show a typical slice if the axis of revolution is the $x$-axis. Then, do the same for the $y$-axis. For each diagram, label the dimensions of the typical slice. Set up and evaluate an integral for each case.

3. What accounts for the difference in volume?

Consider the region bounded by $f(x) = x^3 + 1$, the $x$-axis, and the line $x = 1$.

1. Set up and evaluate the integral that will calculate the volume of the solid created when the region is rotated about the $x$-axis.

2. What if the region is instead rotated about the line $y = -1$? Now what is the shape of the typical slice?

3. For the typical slice from part (b), what is the radius of the inner circle? The outer circle? How is each related to the graph?

4. Calculate the volume of solid created when the region is rotated about the line $y = -1$.

5. Tran is an ace at shifting graphs and has shifted the shaded region from part (b) up one unit and then rotated this new region about the $x$-axis. Should Tran get the same volume as in part (d)? Set up and evaluate an integral to verify your prediction.

1. $\int _ { 2 } ^ { 4 } \frac { 1 } { 9 - 2 x } d x$ 

2. $\int _ { - 3 } ^ { 3 } ( 2 x ^ { 5 } + 3 x ^ { 2 } + 1 ) d x$ 

3. $\int _ { 3 } ^ { 2 } \frac { x ^ { 2 } - 1 } { x - 1 } d x$ 

4. $\int _ { 1 } ^ { 4 } \sqrt { 16 x ^ { 3 } } d x$ 

5. $\int _ { \pi / 4 } ^ { \pi / 2 } \operatorname { sin } ^ { 3 } ( x ) \operatorname { cos } ( x ) d x$ 

1. $\frac { d } { d x } ( \frac { 6 x - 2 } { 3 x + 1 } )$ 

2. $\frac { d } { d x } [ \operatorname { sec } ( 5 \operatorname { ln } [ x ] ) ]$ 

3. $\frac { d } { d x } ( \frac { \operatorname { sin } ( 2 x ) } { x ^ { 3 } } )$ 

4. $\frac { d } { d k } ( k ^ { 2 } 9 ^ { 3 k - 1 } )$ 

5. $\frac { d } { d x } [ \operatorname { sin } ^ { 2 } ( x ) \operatorname { cos } ( x ) ]$ 

The Pi hotel is in the town of Accelerton. The hotel features a grand fireworks display every 4th of July. After launching a firework from the top of the building, the projectile reaches its maximum height where it explodes, amazing the crowds on the ground. The function that models the height of the shell for the firework (in feet) at any time $t$ (in seconds) is given by the function $s(t) = -16t^2 + 120t + 136$. Homework Help ✎

1. How tall is the hotel?

2. What is the initial velocity of the projectile?

3. In order to determine the length of the fuse, the organizers need to know when the projectile will be at its maximum height. At what time should the firework explode?

4. How high will the explosion occur?

5. Oh no! The timing device failed! The shell is falling towards the ground. If the shell is traveling faster than $150$ ft/sec it will explode on contact with the ground. Will the shell explode?

Given the graph of $y = f (x)$ at right, determine the values of $k$ that will make each integral expression equivalent to $\int _ { 0 } ^ { 2 } f ( x ) d x$. Homework Help ✎