Applications of Integration.

[idontknowhwattodo]

I need help so bad.. I have no clue how to these problems. Any help will be appreciated.
[h=3]Part 1[/h]The type of bread chosen for this special calculus toast isn't the square sandwich shape, but the kind that is curved across the top. Imagine that the toast is composed of the curved part sitting atop the rectangular portion. The equation of the curved part of the toast is x²/4 + y² = 1, and it sits directly and perfectly on top of a rectangle of height 3 inches.
a) What are the equations of the rectangular boundaries?
b) Graph the toast boundaries, making certain to include screen shots of the boundary equations, Window settings, and the graph.
c) How would you find the length of the curved part of the toast? What is its numerical approximation based on the calculator's built-in capabilities? Make certain to include a screen shot of the calculated length.
d) Explain and illustrate how you would use calculus to find analytically the areas of the curved part and the rectangular part of the toast.
e) Exactly how much toast area must you cover if you spread peanut butter on the top of the toast?
[h=3]Part 2[/h]Even calculus English muffins are circular and have an approximate diameter of 3.5 inches. Position an English muffin half on the coordinate axes where its center is at the origin.
a) What is the equation of the English muffin half?
b) Create a graph of the English muffin half, making certain that it actually looks circular. Include screen shots of the graph and the equation(s).
c) Explain and illustrate how to use calculus to compute analytically the area of the English muffin half.
d) Exactly how much area must you cover if you spread peanut butter on top of the English muffin half.
e) If you are running low on peanut butter, justify whether you should you have toast or an English muffin half.
[h=3][/h][h=3]Part 3[/h]The special calculus doughnut has the same outside diameter as the English muffin and should be placed on the coordinate axis where its center is at the origin. The diameter of the doughnut hole is 1 inch.
a) Explain and illustrate how to find the exact volume of the doughnut hole that is created by revolving around the x-axis the circular area cut from the dough.
b) Explain how to find the equation of the circular disk that must be revolved around the y-axis to generate this special doughnut.
c) Set up but do not compute the volume the doughnut generated by revolving the circular sector from b) around the y-axis.

 

[Ishuda]

I need help so bad.. I have no clue how to these problems. Any help will be appreciated.
Part 1

The type of bread chosen for this special calculus toast isn't the square sandwich shape, but the kind that is curved across the top. Imagine that the toast is composed of the curved part sitting atop the rectangular portion. The equation of the curved part of the toast is x²/4 + y² = 1, and it sits directly and perfectly on top of a rectangle of height 3 inches.
a) What are the equations of the rectangular boundaries?
b) Graph the toast boundaries, making certain to include screen shots of the boundary equations, Window settings, and the graph.
c) How would you find the length of the curved part of the toast? What is its numerical approximation based on the calculator's built-in capabilities? Make certain to include a screen shot of the calculated length.
d) Explain and illustrate how you would use calculus to find analytically the areas of the curved part and the rectangular part of the toast.
e) Exactly how much toast area must you cover if you spread peanut butter on the top of the toast?
...

You need to turn the problem statements/questions into mathematical statements/questions. For example
(1) The equation of the curved part of the toast is x²/4 + y² = 1, and it sits directly and perfectly on top of a rectangle of height 3 inches. First we note that the curved part is an equation for an ellipse which is left/right symmetric about x = 0 and top/bottom symmetric about y=0. The min/max of x is -2 and 2 and the min/max for y is -1, 1. The upper part of the ellipse is given by
y = \(\displaystyle \frac{1}{2}\sqrt{4 - x^2}\)
I would interpret "sits directly and perfectly on top of a rectangle of height 3 inches" as the top part of the bread is the ellipse intersecting the sides of the bread on the horizontal axis at x = \(\displaystyle \pm\)2. Thus the width of the rectangular piece of bread is 4 inches and it extends 3 inches below the x axis.
NOTE: I would rather have had the problem give the equation for the ellipse as
x²/4+(y-3)² = 1
since that would lead to a 'more natural', IMO, base of the rectangle as the x axis between \(\displaystyle \pm\)2. The above interpretation of "sits directly ..." results for the max/mins of the ellipse x and y values whereas I first thought of an intersection of y=3 for the rectangle and ellipse.

For the remainder of this and the other problems, please show us what you have done so we can know where to help you.

 

[stapel]

I need help so bad.. I have no clue how to these problems. Any help will be appreciated.

You should have at least some clue, since much of the set-up is just algebra.

Part 1:
The type of bread chosen for this special calculus toast isn't the square sandwich shape, but the kind that is curved across the top. Imagine that the toast is composed of the curved part sitting atop the rectangular portion. The equation of the curved part of the toast is x2/4 + y2 = 1, and it sits directly and perfectly on top of a rectangle of height 3 inches.
a) What are the equations of the rectangular boundaries?

What is the graph of the curve? Isn't it an ellipse? What is its center? What is its width? What then is the width of the rectangle?

Draw the rectangle beneath the graph of the top half of the ellipse. Given the height of the rectangle, what must be the equation of the bottom of the piece of toast? (Hint: "y = -3".) What are the equations of the vertical lines for the sides of the piece of toast? (Hint: "x = [x-intercept values]".) The fourth boundary of the rectangle is of course just the x-axis.

b) Graph the toast boundaries, making certain to include screen shots of the boundary equations, Window settings, and the graph.

To learn how to use your graphing calculator, review the owners manual, or consult online resources such as the manufacturer's website.

c) How would you find the length of the curved part of the toast?

What information has your class covered for the "circumferences" of ellipses?

What is its numerical approximation based on the calculator's built-in capabilities? Make certain to include a screen shot of the calculated length.

Check your owner's manual.

d) Explain and illustrate how you would use calculus to find analytically the areas of the curved part and the rectangular part of the toast.

Hint: Integrals find areas.

e) Exactly how much toast area must you cover if you spread peanut butter on the top of the toast?

This is asking you to provide the numerical value of the area.

Part 2:
Even calculus English muffins are circular and have an approximate diameter of 3.5 inches. Position an English muffin half on the coordinate axes where its center is at the origin.
a) What is the equation of the English muffin half?

You are told that this is a circle with radius r = (3.5)/2 and center (h, k) = (0, 0). Use the circle-equation formula you learned back in high school.

b) Create a graph of the English muffin half, making certain that it actually looks circular. Include screen shots of the graph and the equation(s).

Consult your owner's manual.

c) Explain and illustrate how to use calculus to compute analytically the area of the English muffin half.

Hint: Integrals give areas.

d) Exactly how much area must you cover if you spread peanut butter on top of the English muffin half.

Provide the numerical value of the area.

e) If you are running low on peanut butter, justify whether you should you have toast or an English muffin half.

Compare the two numerical values. Note which one is smaller.

Part 3:
The special calculus doughnut has the same outside diameter as the English muffin and should be placed on the coordinate axis where its center is at the origin. The diameter of the doughnut hole is 1 inch.
a) Explain and illustrate how to find the exact volume of the doughnut hole that is created by revolving around the x-axis the circular area cut from the dough.

Hint: Volumes of rotation.

b) Explain how to find the equation of the circular disk that must be revolved around the y-axis to generate this special doughnut.

Hint: Use a different method for volumes of rotation, or at least a different set-up.

c) Set up but do not compute the volume the doughnut generated by revolving the circular sector from b) around the y-axis.

Show the actual set-up.