Have I incorrectly set up the integral for this center of gravity/centroid problem? Consider the bounded region R between the graphs of f(x)=x^2 and g(x)=3x. Sketch R, which appropriate labels. Then set up all the integrals required do not evaluate.

Damon Vazquez

Damon Vazquez

Answered question

2022-09-07

Have I incorrectly set up the integral for this center of gravity/centroid problem?
Consider the bounded region R between the graphs of g ( x ) = 3 x and g(x)=3x. Sketch R, which appropriate labels. Then set up all the integrals required do not evaluate.
My solution:
Definitions:
A = a b ( f ( x ) g ( x ) ) d x
M x = y bar = 1 A a b 1 2 ( f ( x ) 2 g ( x ) 2 ) d x
M y = x bar = 1 A a b x ( f ( x ) g ( x ) ) d x
….
A = 0 3 ( x 2 3 x ) d x = 9 2
M x = y bar = 2 9 0 3 1 2 ( ( x 2 ) 2 ( 3 x ) 2 ) d x
M y = x bar = 2 9 0 3 x ( x 2 3 x ) d x

Answer & Explanation

tiepidolu

tiepidolu

Beginner2022-09-08Added 8 answers

The correct integrals should be
A = 0 3 ( 3 x x 2 ) d x = 9 2
M x = 2 9 0 3 x ( 3 x x 2 ) d x
M y = 2 9 0 3 ( 3 x ) 2 ( x 2 ) 2 2 d x
since 3 x > x 2 over (0,3).

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