Definite Integration
Grade 12
Question:
<p><span class="math-tex">\(\lim\limits _{n \rightarrow \infty} \frac{3}{n}\)</span><span class="math-tex">\(\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+. .+\left(3-\frac{1}{n}\right)^2\right\}\)</span> is equal</p>
<p style="display:inline">0</p>
<p style="display:inline">12</p>
<p style="display:inline"><span class="math-tex">\(\frac{19}{3}\)</span></p>
<p style="display:inline">19</p>
Step-by-Step Solution
Key Concept: Transform the limit of a Riemann sum into a definite integral by substituting r/n with x and 1/n with dx.
<p>Given series can be define as <span class="math-tex">$\lim \limits_{n \rightarrow \infty} \frac{3}{n} \sum_{r=0}^{n-1}\left(2+\frac{r}{n}\right)^2$</span><br />
<span class="math-tex">$\frac{\mathrm{r}}{\mathrm{n}} \rightarrow \mathrm{x}, \frac{1}{\mathrm{n}} \rightarrow \mathrm{dx}$</span><br />
<span class="math-tex">$=3 \int\limits^1_0(2+x)^2 d x$</span> = 27 - 8 = 19</p>
Correct Answer: D