Chapter 5: Applications of Integration
Section 5.7: Centroids
Determine the centroid of the region Rx bounded by fx=cosx,gx=sinx,0≤x≤π/4.
The area of Rx is
∫0π/4cosx−sinx ⅆx = 2−1 =
Its centroid is given by
x&conjugate0;=1A∫0π/4x cosx−sinx ⅆx = π 2−442−1
y&conjugate0; = 1A∫0π/4cos2x−sin2x/2 ⅆx = 1+24
Figure 5.7.3(a) Region Rx and its centroid
The centroid, at approximately 0.267,0.604, is shown as the black dot in Figure 5.7.3(a).
Calculate A, the area of region Rx
Expression palette: Definite-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻A
∫0π/4cosx−sinx ⅆx = 2−1→assign to a nameA
Context Panel: Approximate≻10 (digits)
1A∫0π/4x cosx−sinx ⅆx = −1+14⁢2⁢π2−1→at 10 digits0.2673034979
1A∫0π/4cos2x−sin2x/2 ⅆx = 14⁢2−1→at 10 digits0.6035533912
Note that y&conjugate0; becomes 1+2/4 by rationalizing the denominator in Maple's value for the integral:
Control-drag the expression.
Context Panel: Rationalize
142−1= rationalize 14+14⁢2
The Rationalize option in the Context Panel applies the top-level rationalize command.
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