Curve it Like Calculus
Find the area under y = x from x = 0 to x = 2
Answer:
a =2
Explanation:
https://docs.google.com/document/d/1AZKmcITqDKt58y1IQp-wtxOiTSDuuvD8h7NLC5AHf0I/edit?tab=t.0
Charades
What does this scenario represent?
a) Integrating acceleration gives the total distance jogged, and integrating velocity gives the fastest speed.
b) Integrating acceleration gives the speed change during the jog, and integrating velocity gives the final position after jogging forward and backward.
c) Integrating acceleration gives the average speed, and integrating velocity gives the average position.
d) Integrating acceleration gives the top speed, and integrating velocity gives the farthest point from the start.
B. Integrating acceleration gives your speed change during the jog, and integrating velocity gives your final position after jogging forward and backward.
Explanation:
Integrating acceleration over time gives you the change in velocity during your jog. This is because acceleration is the derivative of velocity, so integrating it recovers the total change in velocity as you speed up, slow down, stop, or reverse while jogging.
Integrating velocity over time gives you the displacement after jogging forward and backward. Velocity is the derivative of position, so integrating velocity tells you where you ended up.
Mismatched
Question:
A region is bounded between the curves y = x^2 and y = x from x = 0 to x = 1
Answer/Explanation/Misconception:
https://docs.google.com/document/d/1eW4RsYqwCVFus4AG30ICzC2ClEPu0LyRlKCA7sPzCiw/edit?tab=t.0
What can be determined from analyzing the scenario?
A) The total amount of material needed, by accounting for the size of each layer as you stack them.
B) Only the number of materials used, without considering their individual sizes.
C) The color and finish of the materials, but not the quantity required.
D) The size of the largest object, but not the total material needed.
Correct Answer:
A) The total amount of material needed, by accounting for the size of each layer as you stack them.
Explanation:
In the scenario, building a tower with bricks that change size as you go up is similar to finding the volume of a solid with varying cross sections. Each layer of bricks represents a cross section whose area changes from the bottom to the top of the tower. By accounting for the size of each layer and summing them all together, you determine the total amount of material needed for the entire tower. This mirrors the mathematical process of integrating the area of each cross section to find the total volume of a solid in calculus.
Pictionary
What does this scenario represent?
A. The total money earned is the sum of all sales at every moment.
B. The total money earned is just the sales during the busiest hour.
C. The total money earned is the difference between the most and least customers.
D. The total money earned is the same every hour.
Correct Answer:
A. The total money earned is the sum of all sales at every moment.
Explanation:
The total money collected represents the sum of all sales throughout the day, just like the area under a sales rate curve shows total revenue.
Charades
What does the scenario represent towards the end?
Choices:
A) The direction of acceleration opposes your motion, so you are speeding up.
B) The direction of acceleration matches your motion, resulting in a decrease in speed.
C) The direction of acceleration opposes your motion, so you are slowing down.
D) The direction of acceleration matches your motion, resulting in an increase in speed.
Correct Answer:
C) The direction of acceleration opposes your motion, so you are slowing down.
Explanation:
As you ride the roller coaster, your speed changes depending on the slope of the track:
Acceleration tells us how your speed (velocity) changes over time. The integral of acceleration over a time interval gives you the total change in velocity during that interval.
Set up the integral for the volume of the solid formed by revolving y = x^2 from x = 0 to x = 2 about the x-axis.
Answer and Explanation:
https://docs.google.com/document/d/1OjfBxJLjluN01x_vQszPvqJ7KnPqHAEy4FYNdjOTNLQ/edit?tab=t.0
Set up the integral for the volume of a solid whose base is the region bounded by y = e^(-x), y = 0 , x = 0, and, x = 1 with semicircular cross-sections perpendicular to the x-axis.
Answer and Explanation:
https://docs.google.com/document/d/1Tv3_CNFSNMToMF_yu709uEAm1MK6qTx5oEy2gQuYVF0/edit?tab=t.0
Pictionary
At the end of the scenario, what does the area under the curve represent during that time?
Choices:
A) The cumulative number of people inside the space remains unchanged, regardless of the rate before or after.
B) The space is continuously admitting new guests at a nonzero but unchanging rate.
C) The total number of admissions during that interval is equal to the length of the interval multiplied by the maximum attendance rate.
D) The accumulated total of all arrivals over the interval, accounting for the varying rate at which people enter.
Correct Answer:
D) The accumulated total of all arrivals over the interval, accounting for the varying rate at which people enter.
Explanation:
In the movie theater example, the graph shows how fast people are entering the theater at different times of the day. The height of the curve tells us how many people are coming in each hour, and the bottom line shows the time. If we look at the area under the curve for a certain time period, it tells us the total number of people who entered during that time. This is because the area adds up all the arrivals, even if the rate changes throughout the day.
BONUS(WHOLE CLASS GETS A POINT)
A runner jogs forward with velocity y = cos(t) from t = 0 to t = 3(pi) . How far did they travel?
Options:
a) 0
b)
c) 4
d) 6
Correct Answer:
d) 6
Explanation:
https://docs.google.com/document/d/1Ypo7SnaBRy4mOMGm3wa88eWALVyX2XgoecZp8IrUEl4/edit?tab=t.0
Set up the integral for the volume of the solid formed by revolving the region between y = x^2 and y = 2x from x = 0 to x = 2 about the x-axis.
Answer and Explanation:
https://docs.google.com/document/d/1NKPT_xyvtmUpt6-AeR939MYKXp0ow1kXamBGMRLLZk4/edit?tab=t.0
Set up the integral for the volume of a solid whose base is the region bounded by y = x, y = 2 - x, and y = 0, with cross-sections perpendicular to the y-axis that are equilateral triangles.
Answer and Explanation:
https://docs.google.com/document/d/1gKyFkzC_McqOV5fBVRvfQPTs65z9RvFdAggPdRjH87Q/edit?tab=t.0