Explanation Given: The triangular region with vertices A ( 1 , 1 ) , B ( 3 , 5 ) , and C ( 3 , − 2 ) . Calculation: Consider the triangular region with vertices A ( 1 , 1 ) , B ( 3 , 5 ) , and C ( 3 , − 2 ) . Plot all three points on the same graph paper, the given region is shown in the following graph: Use the Formula for the Volume of a solid with known cross-sectional area, “A solid S with known cross-sectional area A ( x ) , perpendicular to the x- axis at each point on the interval [ a , b ] has volume V = ∫ a b A ( x ) d x ” Compute the equation of straight lines AB and AC. Use the equation of a straight line passing through two points ( x 1 , y 1 ) and ( x 2 , y 2 ) . y − y 1 = ( y 2 − y 1 x 2 − x 1 ) ( x − x 1 ) Equation of line AB will be y − 1 = ( 5 − 1 3 − 1 ) ( x − x 1 ) y − 1 = 2 ( x − 1 ) y = 2 x − 1 Equation of line AC will be, y − 1 = ( − 2 − 1 3 − 1 ) ( x − x 1 ) y − 1 = − 3 2 ( x − 1 ) y − 1 = − 3 2 x + 3 2 y − 1 = − 3 2 x + 5 2 Take a vertical strip perpendicular to the x- axis in the given region. The length of this strip is, a = ( 2 x − 1 ) − ( − 3 2 x + 3 2 ) a = 2 x − 1 + 3 2 x − 3 2 a = 7 2 x − 7 2 a = 7 2 ( x − 1 ) As each cross-section perpendicular to the x- axis is an equilateral triangle, the area of the cross-sectional will be, A ( x ) =

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 6.2, Problem 8PS

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To find:the volume of the solid whose base is the given region.

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Chapter 6.2, Problem 7PS

Chapter 6.2, Problem 9PS

Chapter 6 Solutions

Calculus

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the volume of the solid whose base is the given region.

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Chapter 6 Solutions