Explanation Given information: The shaded area under the curve is given by the equation y = 4 x 2 + 13 x − 9 x 3 + 2 x 2 − 3 x , the line x = 3 and the line x = 5 . The limits for the variable x are given as 3 for a and 5 for b . Formula: ∫ a b d x x = ln | b | − ln | a | a ln x = ln x a ∫ a b x n d x = b n + 1 − a n + 1 n + 1 Calculation: Express the shaded area using the below expression. ∫ a b y d x Substitute 3 for a , 5 for b and 4 x 2 + 13 x − 9 x 3 + 2 x 2 − 3 x for y . ∫ a b y d x = ∫ 3 5 ( 4 x 2 + 13 x − 9 x 3 + 2 x 2 − 3 x ) d x = ∫ 3 5 ( 4 x 2 + 13 x − 9 x 3 + 3 x 2 − x 2 − 3 x ) d x = ∫ 3 5 ( 4 x 2 + 13 x − 9 x 2 ( x + 3 ) − x ( x + 3 ) ) d x = ∫ 3 5 ( 4 x 2 + 13 x − 9 ( x + 3 ) ( x 2 − x ) ) d x ∫ a b y d x = ∫ 3 5 ( 4 x 2 + 13 x − 9 ( x + 3 ) x ( x − 1 ) ) d x Express the integrand as a sum of partial fractions with undetermined coefficients. 4 x 2 + 13 x − 9 ( x + 3 ) x ( x − 1 ) = A x + B ( x − 1 ) + C ( x + 3 ) (1) Simplify the above equation. 4 x 2 + 13 x − 9 ( x + 3 ) x ( x − 1 ) = A ( x − 1 ) ( x + 3 ) + B x ( x + 3 ) + C x ( x − 1 ) ( x + 3 ) x ( x − 1 ) 4 x 2 + 13 x − 9 = [ A ( x 2 + 2 x − 3 ) + B ( x 2 + 3 x ) + C ( x 2 − x ) ] 4 x 2 + 13 x − 9 = [ x 2 ( A + B + C ) + x ( 2 A + 3 B − C ) − 3 A ] (2) Equate the constant term on both the sides of equation (2). A = 3 Equate the coefficient of the x term on both the sides of equation (2). 2 A + 3 B − C = 13 Substitute 3 for A . 3 B − C = 7 (3) Equate the coefficient of the x 2 term on both the sides of equation (2). A + B + C = 4 Substitute 3 for A . B + C = 1 (4) Sum the equations (3) and (4). 4 B = 8 B = 2 Substitute 2 for B in equation (3). C = − 1 Substitute 3 for A , 2 for B and -1 for C in equation (1). 4 x 2 + 13 x − 9 ( x + 3 ) x ( x − 1 ) = 3 x + 2 ( x − 1 ) − 1 ( x + 3 ) Evaluate the integral of integrand as below. ∫ a b y d x = ∫ 3 5 [ 3 x + 2 ( x − 1 ) − 1 ( x + 3 ) ] d x = [ 3 ln | x | + 2 ln | x − 1 | − ln | x + 3 | ] 3 5 = [ 3 ln | 5 | + 2 ln | 5 − 1 | − ln | 5 + 3 | − 3 ln | 3 | − 2 ln | 3 − 1 | + ln | 3 + 3 | ] = [ 3 ln | 5 | + 2 ln | 4 | − ln | 8 | − 3 ln | 3 | − 2 ln | 2 | + ln | 6 | ] Simplify it further as below

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Chapter 8.4, Problem 76E

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The x-coordinate of the centroid of the region in the first quadrant bounded by the line x=3, the curve y=4x2+13x−9x3+2x2−3x and the line x=5.

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Chapter 8.4, Problem 75E

Chapter 8.4, Problem 77E

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The x -coordinate of the centroid of the region in the first quadrant bounded by the line x = 3 , the curve y = 4 x 2 + 13 x − 9 x 3 + 2 x 2 − 3 x and the line x = 5 .

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The x-coordinate of the centroid of the region in the first quadrant bounded by the line x=3, the curve y=4x2+13x−9x3+2x2−3x and the line x=5.

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