Question

how do i prove this by computing the limit of right endpoint approximations? problem below.

Answer 1

\[\int\limits_{0}^{b} x ^{3} dx = b ^{4} / 4 \]

Answer 2

It should set up like this: \[\int\limits_{0}^{b}x^3dx = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{ b }{ n }(0+\frac{ b }{ n }i)^3\]

Answer 3

\[= \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{ b }{ n }(\frac{ b^3 }{ n^3 }i^3)\]
\[= \lim_{n \rightarrow \infty} \sum_{i=1}^{n} (\frac{ b^4 }{ n^4 }i^3)\]
\[= \lim_{n \rightarrow \infty} \frac{ b^4 }{ n^4 } \sum_{i=1}^{n}(i^3)\]

Answer 4

\[\sum_{i=1}^{n}(i^3)= \frac{ n^2(n+1)^2 }{ 4 }\]
\[= \lim_{n \rightarrow \infty} \frac{ b^4 }{ n^4 }\frac{ n^2(n+1)^2 }{ 4 }\]
\[= \lim_{n \rightarrow \infty} \frac{ b^4 }{ n^2 }\frac{ (n+1)^2 }{ 4 }\]

Answer 5

\[= \lim_{n \rightarrow \infty} \frac{ b^4(n+1)^2 }{ 4n^2 }\]
\[= \lim_{n \rightarrow \infty} \frac{ b^4(n^2+2n+1) }{ 4n^2 }\]
\[= \lim_{n \rightarrow \infty} \frac{ b^4n^2+b^42n+b^4 }{ 4n^2 }\]

Answer 6

\[= \lim_{n \rightarrow \infty} \frac{ \frac{b^4n^2}{n^2}+\frac{b^42n}{n^2}+\frac{b^4}{n^2} }{ \frac{4n^2}{n^2} }\]
\[= \lim_{n \rightarrow \infty} \frac{ b^4+\frac{b^42n}{n^2}+\frac{b^4}{n^2} }{ 4 }\]
\[= \frac{b^4}{4}\]