Explanation Given: The given equation is: y = 4 − x 2 and, n = 8 Approach: Trapezoidal Rule: Let f be a continuous function on [ a , b ] and let [ a , b ] be divided into n equal subintervals by the points a = x 0 , x 1 , x 2 , ⋯ , x n = b . Then, by the trapezoidal rule, ∫ a b f ( x ) d x ≈ ( b − a n ) [ 1 2 f ( x 0 ) + f ( x 1 ) + ⋯ + f ( x n − 1 ) + 1 2 f ( x n ) ] . Calculation: Consider the given equation is, y = 4 − x 2 To find the limits of x , take y = 0 , then 0 = 4 − x 2 0 = 4 − x 2 x = ± 2 Therefore, n = 8 , b = 2 , a = − 2 and, b − a n = 2 − ( − 2 ) 8 = 1 2 Then, x i + 1 cab be calculated by x i + 1 = x i + b − a n . The corresponding function values will be calculated as below: i x i f ( x i ) 0 − 2 To determine (b) To find: The area under the semicircle for the given equation y = 4 − x 2 by using the Simpson’s Rule. To determine (c) The techniques were more accurate after compare the result with area found by the formula for the area of the circle.

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

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ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

Publisher: PEARSON+

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Chapter 8.1, Problem 13E

To determine

(a)

To find:

The area under the semicircle for the given equation y=4−x2 by using the trapezoidal rule.

To determine

(b)

To find:

The area under the semicircle for the given equation y=4−x2 by using the Simpson’s Rule.

To determine

(c)

The techniques were more accurate after compare the result with area found by the formula for the area of the circle.

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Chapter 8.1, Problem 12E

Chapter 8.1, Problem 14E

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Chapter 8 Solutions

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

Ch. 8.1 - YOUR TURN Use the trapezoidal rule with n=4 to...Ch. 8.1 - Prob. 2YT Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E

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(a) To find: The area under the semicircle for the given equation y = 4 − x 2 by using the trapezoidal rule.

Solution Summary: The author explains how to find the area under the semicircle for the given equation by using the trapezoidal rule.

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(a)

To find:

The area under the semicircle for the given equation y=4−x2 by using the trapezoidal rule.

(b)

To find:

The area under the semicircle for the given equation y=4−x2 by using the Simpson’s Rule.

(c)

The techniques were more accurate after compare the result with area found by the formula for the area of the circle.

Want to see the full answer?

Chapter 8 Solutions

(a) To find: The area under the semicircle for the given equation y = 4 − x 2 by using the trapezoidal rule.

Solution Summary: The author explains how to find the area under the semicircle for the given equation by using the trapezoidal rule.

Concept explainers

Videos

(a) To find: The area under the semicircle for the given equation y=4−x2 by using the trapezoidal rule.

(b) To find: The area under the semicircle for the given equation y=4−x2 by using the Simpson’s Rule.

(c) The techniques were more accurate after compare the result with area found by the formula for the area of the circle.

Want to see the full answer?

Chapter 8 Solutions

(a)

To find:

The area under the semicircle for the given equation y=4−x2 by using the trapezoidal rule.

(b)

To find:

The area under the semicircle for the given equation y=4−x2 by using the Simpson’s Rule.

(c)

The techniques were more accurate after compare the result with area found by the formula for the area of the circle.