Please help! Calculus:
A museum curator would like to find out more information on 3 artifacts that she wants to replicate for demonstration purposes. Someone had previously done some work on this project. When she saw equations, she knew she needed to contact someone with some experience in calculus. Unfortunately the information is incomplete. Here's the information received:
Object #1: 3 cm base radius, rotating about the y-axis, y=0 and y=-2x^4+6x^3
Object #2: Rotating about the x-axis, cylindrical shells, widest shell has 10 cm diameter, solid except for 1 cm radius inside, x=0 and x=1/5y^2+2
Object #3:
y=1/2, x=-1, x=1, y=5secx, rotating about the x-axis (all measurements are in cm).
The curator wants you to calculate how much of her 1,200 cubic cm of polymer clay has to be used in order to recreate these objects.
After looking at this information, you decide that you're going to have some fun with integration by creating a 4th solid that uses up the remainder of the polymer clay. You'll send it back to the curator to see if she can figure out which one doesn't represent the real artifact.
Process
Find the volume of item #1.
Find the volume of item #2.
Find the volume of item #3.
Calculate the unused portion of polymer clay.
Create an integral that can be used to find a specific volume while identifying the bounds that make this work.
Task 1
Part A Take a picture of a sketch or use drawing or graphing software to draw the area described by object #1. Insert the image.
Part B Take a picture of a sketch or use drawing software to draw the shape described by object #1. Insert the image.
Part C Use descriptive words to describe the shape of object #1. Can you describe the shape as an existing object?
Part D Find the volume of object #1.
Part E Can this solid be found by using disk/washers and cylindrical shells? Explain.
Task 2
Part A Take a picture of a sketch or use drawing or graphing software to draw the area described by object #2. Insert the image.
Part B Take a picture of a sketch or use drawing software to draw the shape described by object #2. Insert the image.
Part C Use descriptive words to describe the shape of object #2. Can you describe the shape as an existing object?
Part D Find the volume of object #2.
Task 3
Part A Take a picture of a sketch or use drawing or graphing software to draw the area described by object #3. Insert the image.
Part B Take a picture of a sketch or use drawing software to draw the shape described by object #3. Insert the image.
Part C Use descriptive words to describe the shape of object #3. Can you describe the shape as an existing object?
Part D Find the volume of object #3.
Part E If this solid was rotated around the line y = -1, how would the integral change? Without calculating the value of the integral, would the volume be more or less? Explain.
Task 4
Part A Compute the total volume of the 3 objects. Determine the remaining amount clay.
Part B Using the information from the previous question, setup an integral that is equal to the volume of the remaining polymer clay. Complete the integration.
Part C Describe the solid you're creating.
Part D Identify the equations and the limits of integration (a and b) used to create this object.
Part E Do you think the curator will be able to pick out the item you created? Explain.
