Investigating ratios of areas and volumes

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November 28th 2008, 03:08 PM
chessmastery2001

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Investigating ratios of areas and volumes

The ratio of the areas formed when y=x^n is graphed between two arbitrary parameters x=a and x=b such that a<b.

1. Given the function y=x^2, consider the region formed by this function from x=0 to x=1 and the x-axis. Label the region from y=0 to y=1 and the y-axis area A.

Attachment 8984

Find the ratio of area A: area B.

Calculate the ratio of the areas for other functions of the type y=x^n, nZ^+ between x=0 and x=1. Make a conjecture and test your conjecture for other subsets of the real numbers.
[IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
2. Does your conjecture hold only for areas between x=0 and x=1? Examine for x=0 and x=2, x=1 and x=2, etc.

Attachment 8985

3. Is your conjecture true for the general case y=x^n from x=a to x=b such that a<b and for the regions defined below? If so prove it; if not, explain why not.

Area A: y=x^n, y=a^n, y=b^n and the y-axis
Area B: y=x^n, x=a, x=b and the x-axis
December 2nd 2008, 09:00 PM
mr fantastic

Quote:

Originally Posted by chessmastery2001

The ratio of the areas formed when y=x^n is graphed between two arbitrary parameters x=a and x=b such that a<b.

1. Given the function y=x^2, consider the region formed by this function from x=0 to x=1 and the x-axis. Label the region from y=0 to y=1 and the y-axis area A.

Attachment 8984

Find the ratio of area A: area B.

Calculate the ratio of the areas for other functions of the type y=x^n, nZ^+ between x=0 and x=1. Make a conjecture and test your conjecture for other subsets of the real numbers.
[IMG]file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
2. Does your conjecture hold only for areas between x=0 and x=1? Examine for x=0 and x=2, x=1 and x=2, etc.

Attachment 8985

3. Is your conjecture true for the general case y=x^n from x=a to x=b such that a<b and for the regions defined below? If so prove it; if not, explain why not.

Area A: y=x^n, y=a^n, y=b^n and the y-axis
Area B: y=x^n, x=a, x=b and the x-axis

1. Area B = $\int_{0}^{1} x^2 \, dx = \, ....$

Area A = 1 - Area B.

Use this as a model for how to answer the other questions.
December 4th 2008, 03:51 AM
chessmastery2001

1 Attachment(s)

Calculus

I need some more help with problems 2-4. I've attached a pdf file. Attachment 9087

Thanks Again!
December 4th 2008, 04:25 AM
mr fantastic

Quote:

Originally Posted by chessmastery2001

I need some more help with problems 2-4. I've attached a pdf file. Attachment 9087

Thanks Again!

If you understand how to do Q1 you should be able to attempt Q2-4. The pdf file looks like an assignment which will be assessed and contribute towards your final grade. Sorry, but help cannot be given under those circumstances.

Thread closed.