Explanation Given: The base of the solid is an isosceles right triangle whose legs are each 4 units long. Each cross section perpendicular to a side is a semicircle. Formula Used: A solid S with known cross-sectional area A(y), perpendicular to the y-axis each point on the interval [ a , b ] has volume V = ∫ a b A ( y ) d y Calculation: The isosceles right triangle AOB is shown as follows: Legs OA and OB of triangle AOB lie on co-ordinate axes. Thus, co-ordinates of point A will be (0.4) and co-ordinates of point B will be (4, 0) so, equation of line AB will be, y − 0 = 4 − 0 0 − 4 ( x − 4 ) y = − 1 ( x − 4 ) − y = x − 4 x = 4 − y Use the formula for the volume of a solid with known cross-sectional area, A solid S with known cross-sectional area A(y), perpendicular to the y-axis each point on the interval [ a , b ] has volume V = ∫ a b A ( y ) d y Take horizontal strip in the region. The length of this strip will be, d = MN = 4 - y Each cross-section perpendicular to the x-axis is a semi-circle and area of a semi-circle with diameter d is as below, 1 2 π ( d 2 ) 2 = 1 8 π d 2 So, the area of the cross-sectional will be, A ( x ) = 1 8 π d 2 = 1 8 π × ( 4 − y ) 2 =

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ISBN: 9781524916817

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

Publisher: Kendall Hunt Publishing

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Chapter 6.2, Problem 48PS

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To find:The volume of the solid.

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Chapter 6.2, Problem 47PS

Chapter 6.2, Problem 49PS

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5.3 problem 54 FIND THE VOLUME OF THE DESCRIBED SOLID S

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Chapter 6 Solutions

Calculus

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The volume of the solid.

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Chapter 6 Solutions