Explanation Given: Show the integral function as follows: ∫ a b ( c x + d ) f ( x ) d x = ( c x ¯ + d ) ∫ a b f ( x ) d x The integral function of left hand side (LHS) is ∫ a b ( c x + d ) f ( x ) d x . Calculation: The ( x ¯ ) coordinate of centroid is expressed as follows: x ¯ = 1 A ∫ a b x f ( x ) d x (1) Rearrange Equation (1). x ¯ A = ∫ a b x f ( x ) d x The area of the function as follows: A = ∫ a b f ( x ) d x The integral function as follows: ∫ a b ( c x + d ) f ( x ) d x = ∫ a b [ c x f ( x ) + d f ( x ) ] d x = ∫ a b c x f ( x ) d x + ∫ a b d f ( x ) d x = c ∫ a b x f ( x ) d x + d ∫ a b f ( x ) d x (2) Substitute x ¯ A for ∫ a b x f ( x ) d x in Equation (2)

8th Edition

ISBN: 9781285741550

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 8.3, Problem 43E

To determine

To prove: The integral function as ∫ab(cx+d)f(x)dx=(cx¯+d)∫abf(x)dx

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 42E

Chapter 8.3, Problem 44E

Students have asked these similar questions

3. (i) Let A, B, C, D, E be positive constants. Find the limit n³ D lim A(BC)³ +1 en 4+E. n→∞ [6 Marks] (ii) Let X be a random variable with the standard normal distribution, i.e. X has the probability density function Let 1 f(x) √2π n3x2 Xn = 100(50 +X6) en³4, n = 1, 2,..., 3+1 (a) Find the limit lim∞ Xn, verify your answer. Y = 100(50 +X) 22. [6 Marks] (b) Is Xn≤Y? Verify your answer. [6 Marks] (c) Is EY∞o? Verify your answer. [6 Marks] (d) Does lim∞ EXn (e) Find lim∞ EXn. Elim Xn? Why? [6 Marks] [3 Marks]

2. Let (, F, P) be a probability space. Let X, Y be two independent ran- dom variables such that 1 3 P(X = 0) P(X = 1) P(X = 2) P(X = 3) P(X = 4) 10 10 5' 3 P(Y = 0) P(Y = 1) 10 1 10 7 1 P(Y2) = P(Y = 3) , P(Y = 4) 10 20 20 (i) Find the probability generation function Gx(s) and Gy(s) of X and Y, respectively. [12 Marks] (ii) Using Gx(s) and Gy(s), find the mean and variance of X and Y. [12 Marks] (iii) Find the probability generating function Gx+y(s) of X + Y. [8 Marks]

3. (i) Using the definition of the line integral of a vector field, calculate the line integral L³ F.dy of the vector field F: R² → R² given by F(x, y) = (y, x), and where the curve & is the unit semi-circle centred at the origin, located in the upper half-plane and oriented in the anticlockwise direction. Hint. Represent the curve y as the join of two curves y = 71 + 1/2 (see Example 8.9 in the Notes). [20 Marks] (ii) Calculate the same integral using Green's Theorem. [10 Marks]

Chapter 8 Solutions

Calculus: Early Transcendentals

Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Use the arc length formula to find the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To prove: The integral function as ∫ a b ( c x + d ) f ( x ) d x = ( c x ¯ + d ) ∫ a b f ( x ) d x

Solution Summary: The author explains how the integral function of left hand side (LHS) is proved.

Videos

To prove: The integral function as ∫ab(cx+d)f(x)dx=(cx¯+d)∫abf(x)dx

Want to see the full answer?

Chapter 8 Solutions