# Investigating ratios of areas and volumes

• Nov 28th 2008, 03:08 PM
chessmastery2001
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.
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
• Dec 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.
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 = $\displaystyle \int_{0}^{1} x^2 \, dx = \, ....$

Area A = 1 - Area B.

Use this as a model for how to answer the other questions.
• Dec 4th 2008, 03:51 AM
chessmastery2001
Calculus
I need some more help with problems 2-4. I've attached a pdf file. Attachment 9087

Thanks Again!
• Dec 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.