1. ## Hello!

Let b>a>0, k be any real number except -1. Show that the area A of the region bounded by y=x^k, Y=0,x=a,x=b is A=[b^(k+1)-a^(k+1)]/(k+1) square units

2. ## Re: Hello!

Have you tried evaluating \displaystyle \begin{align*} \int_a^b{x^k\,\mathrm{d}x} \end{align*}?

3. ## Re: Hello!

i know. But i have to be compeled different way. In solution, Let t=(b/a)^(1/n)b and x0=a, x1=at, x2=a(t^2),..., xn=a(t^n)=b
Use Area of R= lim Sn (n to infinite) , Delta xi=(b-a)/n and xi=a+i*delta x
Sn= f(x1)*delta(x1)+ . . .f(xn)*delta(xn). YOU KNOW THIS WAY

4. ## Re: Hello!

Oh boy it will be a nightmare if k is negative. When k is positive, though, you should look up the summation formula for sigma (i=1~n) i^k for any positive exponent k.
Example:
sigma (i=1~n) i^2 = n(n+1)(2n+1)/6.