Explanation 1) Concept: The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles A = lim n → ∞ ⁡ R n = l i m n → ∞ f x 1 ∆ x + f x 2 ∆ x + … + f x n ∆ x = l i m n → ∞ ∑ i = 1 n f x i ∆ x The width of the interval a , b is b - a , so the width of each n strip is ∆ x = b - a n where x 0 = a and x n = b . The right endpoints of the subintervals are x i = a + i ∆ x . 2) Given: y = x 4 + 5 x 2 + x , 2 ≤ x ≤ 7 3) Calculation: It is given that y = x 4 + 5 x 2 + x Here a = 2 , b = 7 So ∆ x = b - a n = 7 - 2 n ∆ x = 5 n Now find out x i = a + i ∆ x x 1 = 2 + 1 5 n = 2 + 5 n x 2 = 2 + 2 5 n = 2 + 10 n x 3 = 2 + 3 5 n To determine b) To evaluate: The sum in the expression of the area under the curve To determine c) To find: The exact area by evaluating the limit

8th Edition

ISBN: 9781337051545

Author: Stewart

Publisher: CENGAGE C

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Chapter 4.1, Problem 30E

To determine

a)

To express:

The area under the curve

To determine

b)

To evaluate:

The sum in the expression of the area under the curve

To determine

c)

To find:

The exact area by evaluating the limit

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Chapter 4.1, Problem 29E

Chapter 4.1, Problem 31E

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Chapter 4 Solutions

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a) To express: The area under the curve

Solution Summary: The author explains the area A of the region S that lies under the graph of a continuous function f.

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a) To express: The area under the curve

b) To evaluate: The sum in the expression of the area under the curve

c) To find: The exact area by evaluating the limit

Want to see the full answer?

Chapter 4 Solutions

a)

To express:

The area under the curve

b)

To evaluate:

The sum in the expression of the area under the curve

c)

To find:

The exact area by evaluating the limit