Question

The value of
\[ \int_0^1 a^k x^k\,dx \] is

• A. \(\displaystyle \lim_{n\to\infty}\frac{a^k(1^k+2^k+3^k+\cdots+n^k)}{n^{k+1}}\)
• B. \(\displaystyle \lim_{n\to\infty}\frac{a^k+a^k+\cdots+a^k}{n^{k+1}}\)
• C. \(\displaystyle \lim_{n\to\infty}\frac1n\sum \left(\frac{r}{n}\right)^k\)
• D. \(\displaystyle \lim_{n\to\infty}\frac1n\sum \left(\frac{2r}{n}\right)^k\)

Hint:

To convert a definite integral into a limit of sum over \([0,1]\), use \(\displaystyle \int_0^1 f(x)\,dx=\lim_{n\to\infty}\frac1n\sum_{r=1}^{n}f\left(\frac{r}{n}\right)\).

Answer 1

The Correct Option is A

Step 1: Write the integral as a limit of sum.
For \(\displaystyle \int_0^1 f(x)\,dx\), the limit form is
\[ \int_0^1 f(x)\,dx = \lim_{n\to\infty}\frac1n\sum_{r=1}^{n}f\left(\frac{r}{n}\right) \]

Step 2: Substitute \(f(x)=a^kx^k\).
Here,
\[ f(x)=a^kx^k \]
Therefore,
\[ \int_0^1 a^kx^k\,dx = \lim_{n\to\infty}\frac1n\sum_{r=1}^{n}a^k\left(\frac{r}{n}\right)^k \]

Step 3: Simplify the summation.
\[ = \lim_{n\to\infty}\frac1n\sum_{r=1}^{n}\frac{a^kr^k}{n^k} \]
\[ = \lim_{n\to\infty}\frac{a^k}{n^{k+1}}\sum_{r=1}^{n}r^k \]
\[ = \lim_{n\to\infty}\frac{a^k(1^k+2^k+3^k+\cdots+n^k)}{n^{k+1}} \]

Step 4: Final conclusion.
Hence,
\[ \boxed{\displaystyle \lim_{n\to\infty}\frac{a^k(1^k+2^k+3^k+\cdots+n^k)}{n^{k+1}}} \]