Explanation Given: y = x , y = x 2 , ρ = k x y Formula Used: m = ∫ x 1 x 2 ∫ y 1 y 2 ρ ( x , y ) d y d x M x = ∫ x 1 x 2 ∫ y 1 y 2 y ⋅ ρ ( x , y ) d y d x M y = ∫ x 1 x 2 ∫ y 1 y 2 x ⋅ ρ ( x , y ) d y d x ( x ¯ , y ¯ ) = ( M y m , M x m ) I x = ∫ x 1 x 2 ∫ y 1 y 2 y 2 ⋅ ρ ( x , y ) d y d x I y = ∫ x 1 x 2 ∫ y 1 y 2 x 2 ⋅ ρ ( x , y ) d y d x I 0 = ∫ x 1 x 2 ∫ y 1 y 2 ( x 2 + y 2 ) ⋅ ρ ( x , y ) d y d x x ¯ ¯ = I y m y ¯ ¯ = I x m Where ( x 1 , x 2 ) and ( y 1 , y 2 ) are the limits of the integration. Calculation: First, draw the graph for the given equations by performing the following steps: 1. Draw the reference axis in xy coordinates. 2. Plot the given equations: y = x , y = x 2 3. Mark the points on the plot and connect the points. 4. The area of interest is the area inscribed by the equations, so shade the inside area of the graph. 5. The final plot is as shown below:

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Chapter 14.4, Problem 34E

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To calculate: The Moments of Inertia and Radii of Gyration.

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Chapter 14.4, Problem 33E

Chapter 14.4, Problem 35E

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Chapter 14 Solutions

EBK CALCULUS: EARLY TRANSCENDENTAL FUNC

Ch. 14.1 - Evaluate the iterated integral: 0433cosrdrd Ch. 14.1 - Prob. 1E Ch. 14.1 - Evaluate the integral: xx2yxdy Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Evaluate the integral: 04x2x2ydy Ch. 14.1 - Evaluate the integral: x3x(x2+3y2)dy Ch. 14.1 - Evaluate the integral: eyyylnxxdx;y0 Ch. 14.1 - Evaluate the integral: 1y21y2(x2+y2)dx Ch. 14.1 - Evaluate the integral: 0x2yeyxdy

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The Moments of Inertia and Radii of Gyration.

Solution Summary: The author explains how to calculate the Moments of Inertia and Radii of Gyration.

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To calculate: The Moments of Inertia and Radii of Gyration.

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Chapter 14 Solutions