Determine a region whose area is equal to the given limit. Do not evaluate the limit.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{3}{n} \sqrt{1 +\frac{3i}{n}} $

Determine a region whose area is equal to the given limit. Do not evaluate the limit.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{\pi}{4n} \tan{\frac{i \pi}{4n}} $

(a) Use Definition 2 to find an expression for the area under the curve $ y = x^3 $ from 0 to 1 as a limit.

(b) The following formula for the sum of the cubes of the first $ n $ integers is proved in Appendix E. Use it to evaluate the limit in part (a).

$$ 1^3 + 2^3 + 3^3 + \cdots + n^3 = \biggl[ \frac{n(n + 1)}{2} \biggr]^2 $$

Let $ A $ be the area under the graph of an increasing continuous function $ f $ from $ a $ to $ b $, and let $ L_n $ and $ R_n $ be the approximations to $ A $ with $ n $ subintervals using left and right endpoints, respectively.

(a) How are $ A $, $ L_n $, and $ R_n $ related?

(b) Show that $$ R_n - L_n = \frac{b - a}{n} [f(b) - f(a)] $$
Then draw a diagram to illustrate this equation by showing that the $ n $ rectangles representing $ R_n - L_n $ can be reassembled to form a single rectangle whose area is the right side of the equation.

(c) Deduce that $$ R_n - A < \frac{b - a}{n} [f(b) - f(a)] $$

If $ A $ is the area under the curve $ y = e^x $ from 1 to 3, use Exercise 27 to find a value of $ n $ such that $ R_n - A < 0.0001 $.