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Problem 28

If $ A $ is the area under the curve $ y = e^x $ …

01:04
Problem 28

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 $.
Problem 29

(a) Express the area under the curve $ y = x^5 $ from 0 to 2 as a limit.

(b) Use a computer algebra system to find the sum in your expression from part (a).

(c) Evaluate the limit in part (a).
Problem 30

Find the exact area of the region under the graph of $ y = e^{-x} $ from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b).
Problem 31

Find the exact area under the cosine curve $ y = \cos x $ from $ x = 0 $ to $ x = b $, where $ 0 \le b \le \pi/2 $. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if $ b = \pi/2 $?
Problem 32

(a) Let $ A_n $ be the area of a polygon with $ n $ equal sides inscribed in a circle with radius $ r $. By dividing the polygon into $ n $ congruent triangles with central angle $ 2\pi/n $, show that
$$ A_n = \frac{1}{2} nr^2 \sin \biggl( \frac{2 \pi}{n} \biggr) $$

(b) Show that $ \displaystyle \lim_{n \to \infty} A_n = \pi r^2 $. [$ Hint: $ Use Equation 3.3.2 on page 191.]
FL
Frank L.
Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Problem 15 Problem 16 Problem 17 Problem 18 Problem 19 Problem 20 Problem 21 Problem 22 Problem 23 Problem 24 Problem 25 Problem 26 Problem 27 Problem 28 Problem 29 Problem 30 Problem 31 Problem 32
Problem 27

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

Answer

(A). $L_{n} \leq A \leq R_{n}$
(B). See explanation for result.
(C). $R_{n}-A<\frac{b-a}{n}[f(b)-f(a)]$

Chapter 5
Integrals
Section 1
Areas and Distances
Calculus: Early Transcendentals

Topics

Areas and Distances

Discussion

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Video Transcript

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