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Chapter 9: Q. 14. (page 755)

Give a geometric explanation of why

鈭�02蟺nr2d胃=蟺r2

for any positive real number r and any positive integer nWould the equation also hold for non-integer values of n?

Short Answer

Expert verified

The equation鈭�02蟺nr2d胃=蟺r2 holds true for any non-integer values ofn

Step by step solution

01

Given information

Now consider integral鈭�02蟺nr2d胃

02

The objective is to find to prove that the integral is equal to πr2 

蟺r2for non-integer values of n

Take the integral now.

n2鈭�02蟺nr2d胃=nr22鈭�02蟺n1d胃

03

Prove that any positive real number r and any positive integer  Would the equation also hold for non-integer values of n?

If ris a positive real number and nis a non-negative integer, then That is, both nand rare constants. As a result, they are removed, and the remaining value is integrated within the stated limitations.

鈭�02蟺nr2d胃=nr22胃2=nr222蟺n-0=蟺r2

Hence, the equation holds true for any non-integer values ofn

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