5.43 and 5.44
Determine by direct integration the centroid of the area shown. Express your answer in terms of a and b.
Fig. P5.43
The centroid of shaded area in Fig. P5.43 by method of direct integration.
Answer to Problem 5.43P
Centroid is located at
Explanation of Solution
Refer the figure P5.43 and figure given below.
Write the equation for curve
Here,
Consider the point
Here,
Rewrite equation (I) by substituting
Rewrite the above equation in terms of
Rewrite equation (I) by substituting the above relation for
Rewrite the above equation in terms of
Divide the shaded region in P5.43 into two parts for the purpose of integration. Region
Consider the region
Consider a rectangular differential area element in the region. Write the expression for the x-coordinate of center of mass of differential area element.
Here,
Write the expression for the y-coordinate of center of mass of differential area element in region
Here,
Rewrite the above relation by substituting
Write the expression to calculate the differential area element in
Here,
Rewrite the above relation by substituting
Consider the region
Write the expression for the x-coordinate of center of mass of differential area element in region
Here,
Write the expression for the y-coordinate of center of mass of differential area element in region
Here,
Calculate the slope of
Here,
Write the equation of
Rewrite the above equation by substituting
Rewrite equation (IV) by substituting the above relation for
Write the expression for
Rewrite the above relation by substituting
Write the equation to calculate the total area of shaded region in P5.43.
Here,
Rewrite the above equation by substituting equation (III) and (V).
Write the expression for
Rewrite the above equation by substituting equation (III) and (V).
Write the expression for
Rewrite the above equation by substituting equations (II), (III), (V) and (VI).
Write the expression for first moment of whole area about y-axis.
Here,
Rewrite the above relation by substituting
Rewrite the above relation in terms of
Write the expression for first moment of whole area about x-axis.
Here,
Rewrite the above relation in terms of
Therefore, the centroid is located at
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