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Page 2 Free coa ote Prove that the volume of a hemisphere is a Prove that the volume of a sphere of ra ius a is a segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the volume of the segment is / of the volume of the sphere x in the volume of the para oloi generate y the revolution of the para ola y ax a out x axis from x to x h Sol he equation of the para ola is y ax he require volume of the para oloi is ? y x ? ax x a * x + h x he area of the para ola y ax lying etween the vertex an the latus re tum is revolve a out x axis in the volume generate Sol he equation of the para ola is y ax y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi he limits for the para oloi will e from x to x a en e the require volume of the para oloi is ? y x ? ax x a * x + a a a x in the volume of the soli generate y revolving the ellipse x a y a out the x axis Sol the equation of the ellipse is x a y a out the x axis he require volume of the soli ? y x ? a (a x ) x * x a y + a *a x x + a x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out the major axis Sol he equation of the ellipse is x y n the equation of the auxiliary ir le is x y a the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is say hen a n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is hen a he volume of the soli generate y the revolution of the ellipse a out y axis is ? x y ? a ( y ) y * x a y + Page 3 Free coa ote Prove that the volume of a hemisphere is a Prove that the volume of a sphere of ra ius a is a segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the volume of the segment is / of the volume of the sphere x in the volume of the para oloi generate y the revolution of the para ola y ax a out x axis from x to x h Sol he equation of the para ola is y ax he require volume of the para oloi is ? y x ? ax x a * x + h x he area of the para ola y ax lying etween the vertex an the latus re tum is revolve a out x axis in the volume generate Sol he equation of the para ola is y ax y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi he limits for the para oloi will e from x to x a en e the require volume of the para oloi is ? y x ? ax x a * x + a a a x in the volume of the soli generate y revolving the ellipse x a y a out the x axis Sol the equation of the ellipse is x a y a out the x axis he require volume of the soli ? y x ? a (a x ) x * x a y + a *a x x + a x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out the major axis Sol he equation of the ellipse is x y n the equation of the auxiliary ir le is x y a the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is say hen a n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is hen a he volume of the soli generate y the revolution of the ellipse a out y axis is ? x y ? a ( y ) y * x a y + Free M a ? ( y ) y a * y y + a (say) ow the mean proportional etween an v( ) v a ( a ) a olume generate when ellipse is revolve a out major axis x he urve y (a x) x ( a x) revolves a out the x axis in the volume generate y the loop Sol he given urve is y (a x) x ( a x) he require volume generate y the loop ? y y ? x ( a x) (a x) x ? , x ax a a x a - x * x a x a x a log(x a) + a , log - x in the volume generate y the revolution of the loop of the urve y (a x) x (a x) a out the x axis Sol he given urve is y (a x) x (a x) he require volume generate y the loop ? y x ? x (a x) a x x ? , x ax a a a x - x * x a x a x a log(a x)+ * a a a a log a a loga+ * a a log + [log ] x in the volume of the soli generate y the revolution of the loop of the urve y x (a x) a out the x axis Sol he given equation is y x (a x) ? y x ? x (a x) x * ax x + a [ ] a x in the volume of the soli generate y the revolution of the urve y a (a x ) a out its asymptote Sol he given equation of the urve is y a (a x ) or x y a (a y) he require volume ? y x ? a (x a ) x Putting x atan x ase he require volume a ? se se / a ? os / a ( erify ) x Prove that the volume of the soli generate y revolving the asteroi x y a a out the x axis is a Solution he given urve is x / y / a / he require volume ? y x ? (a / x / ) x Page 4 Free coa ote Prove that the volume of a hemisphere is a Prove that the volume of a sphere of ra ius a is a segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the volume of the segment is / of the volume of the sphere x in the volume of the para oloi generate y the revolution of the para ola y ax a out x axis from x to x h Sol he equation of the para ola is y ax he require volume of the para oloi is ? y x ? ax x a * x + h x he area of the para ola y ax lying etween the vertex an the latus re tum is revolve a out x axis in the volume generate Sol he equation of the para ola is y ax y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi he limits for the para oloi will e from x to x a en e the require volume of the para oloi is ? y x ? ax x a * x + a a a x in the volume of the soli generate y revolving the ellipse x a y a out the x axis Sol the equation of the ellipse is x a y a out the x axis he require volume of the soli ? y x ? a (a x ) x * x a y + a *a x x + a x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out the major axis Sol he equation of the ellipse is x y n the equation of the auxiliary ir le is x y a the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is say hen a n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is hen a he volume of the soli generate y the revolution of the ellipse a out y axis is ? x y ? a ( y ) y * x a y + Free M a ? ( y ) y a * y y + a (say) ow the mean proportional etween an v( ) v a ( a ) a olume generate when ellipse is revolve a out major axis x he urve y (a x) x ( a x) revolves a out the x axis in the volume generate y the loop Sol he given urve is y (a x) x ( a x) he require volume generate y the loop ? y y ? x ( a x) (a x) x ? , x ax a a x a - x * x a x a x a log(x a) + a , log - x in the volume generate y the revolution of the loop of the urve y (a x) x (a x) a out the x axis Sol he given urve is y (a x) x (a x) he require volume generate y the loop ? y x ? x (a x) a x x ? , x ax a a a x - x * x a x a x a log(a x)+ * a a a a log a a loga+ * a a log + [log ] x in the volume of the soli generate y the revolution of the loop of the urve y x (a x) a out the x axis Sol he given equation is y x (a x) ? y x ? x (a x) x * ax x + a [ ] a x in the volume of the soli generate y the revolution of the urve y a (a x ) a out its asymptote Sol he given equation of the urve is y a (a x ) or x y a (a y) he require volume ? y x ? a (x a ) x Putting x atan x ase he require volume a ? se se / a ? os / a ( erify ) x Prove that the volume of the soli generate y revolving the asteroi x y a a out the x axis is a Solution he given urve is x / y / a / he require volume ? y x ? (a / x / ) x Free c Putting x asin x a sin os ? a os a sin os a ? sin os a . / ( ) . / a x Show that the volume of the soli generate y the revolution of the urve (a x)y a x a out its asymptote is a Solution he given urve is (a x)y a x he require volume ? (a x) y ? *a ay a y + y ? a (a y ) y a ? y (a y ) Put y a tan y ase he require volume a ? ase a ( tan ) / a ? se / a ? os / a a x in the volume of the soli generate y the revolution of the issoi y ( a x) x a out its asymptote Solution he given urve is y ( a x) x he require volume ? ( a x) y ? ( a x) x ow y ( a x) x or y v( x a x ) y x ( a x) / x / ( a x) ( a x) x / ( a x) x / ( a x) / y x x / ( a x) ( a x) / y x / ( a x) ( a x) / x en e the require volume ? ( a x) x / ( a x) ( a x) / x ? ( a x) v( ax x ) x ? ( a a sin ) a sin os a sin os (Put x a sin x a sin os ) a ? ( sin ) sin os a *? sin os ? sin os + a * ( / ) ( / ) ( ) ( / ) ( / ) ( ) + a [ ] a x in the volume of the soli s forme y revolving the y loi x a( sin ) y a ( os ) a out its ase Sol he given equations are x a( sin ) y a( os ) en e the require volume ? y x ? a ( os ) x ? a ( os ) a( os ) Page 5 Free coa ote Prove that the volume of a hemisphere is a Prove that the volume of a sphere of ra ius a is a segment is ut off from a sphere of ra ius a y a plane at a istan e a / from the entre Show that the volume of the segment is / of the volume of the sphere x in the volume of the para oloi generate y the revolution of the para ola y ax a out x axis from x to x h Sol he equation of the para ola is y ax he require volume of the para oloi is ? y x ? ax x a * x + h x he area of the para ola y ax lying etween the vertex an the latus re tum is revolve a out x axis in the volume generate Sol he equation of the para ola is y ax y revolving the area of the given para ola lying etween the vertex an the latus retum we get a para oloi he limits for the para oloi will e from x to x a en e the require volume of the para oloi is ? y x ? ax x a * x + a a a x in the volume of the soli generate y revolving the ellipse x a y a out the x axis Sol the equation of the ellipse is x a y a out the x axis he require volume of the soli ? y x ? a (a x ) x * x a y + a *a x x + a x Prove that the volume of the soli gererate y the revolution of an ellipse roun its minor axis is a means proportional etween those generate y the revolution of the ellipse an of the auxliary ir le a out the major axis Sol he equation of the ellipse is x y n the equation of the auxiliary ir le is x y a the volume of the soli generate y the revolution of the ellipse a out the major axis (x axis) is say hen a n the volume of the sphere generate y the revolution of the auxliary ir le a out x axis is hen a he volume of the soli generate y the revolution of the ellipse a out y axis is ? x y ? a ( y ) y * x a y + Free M a ? ( y ) y a * y y + a (say) ow the mean proportional etween an v( ) v a ( a ) a olume generate when ellipse is revolve a out major axis x he urve y (a x) x ( a x) revolves a out the x axis in the volume generate y the loop Sol he given urve is y (a x) x ( a x) he require volume generate y the loop ? y y ? x ( a x) (a x) x ? , x ax a a x a - x * x a x a x a log(x a) + a , log - x in the volume generate y the revolution of the loop of the urve y (a x) x (a x) a out the x axis Sol he given urve is y (a x) x (a x) he require volume generate y the loop ? y x ? x (a x) a x x ? , x ax a a a x - x * x a x a x a log(a x)+ * a a a a log a a loga+ * a a log + [log ] x in the volume of the soli generate y the revolution of the loop of the urve y x (a x) a out the x axis Sol he given equation is y x (a x) ? y x ? x (a x) x * ax x + a [ ] a x in the volume of the soli generate y the revolution of the urve y a (a x ) a out its asymptote Sol he given equation of the urve is y a (a x ) or x y a (a y) he require volume ? y x ? a (x a ) x Putting x atan x ase he require volume a ? se se / a ? os / a ( erify ) x Prove that the volume of the soli generate y revolving the asteroi x y a a out the x axis is a Solution he given urve is x / y / a / he require volume ? y x ? (a / x / ) x Free c Putting x asin x a sin os ? a os a sin os a ? sin os a . / ( ) . / a x Show that the volume of the soli generate y the revolution of the urve (a x)y a x a out its asymptote is a Solution he given urve is (a x)y a x he require volume ? (a x) y ? *a ay a y + y ? a (a y ) y a ? y (a y ) Put y a tan y ase he require volume a ? ase a ( tan ) / a ? se / a ? os / a a x in the volume of the soli generate y the revolution of the issoi y ( a x) x a out its asymptote Solution he given urve is y ( a x) x he require volume ? ( a x) y ? ( a x) x ow y ( a x) x or y v( x a x ) y x ( a x) / x / ( a x) ( a x) x / ( a x) x / ( a x) / y x x / ( a x) ( a x) / y x / ( a x) ( a x) / x en e the require volume ? ( a x) x / ( a x) ( a x) / x ? ( a x) v( ax x ) x ? ( a a sin ) a sin os a sin os (Put x a sin x a sin os ) a ? ( sin ) sin os a *? sin os ? sin os + a * ( / ) ( / ) ( ) ( / ) ( / ) ( ) + a [ ] a x in the volume of the soli s forme y revolving the y loi x a( sin ) y a ( os ) a out its ase Sol he given equations are x a( sin ) y a( os ) en e the require volume ? y x ? a ( os ) x ? a ( os ) a( os ) Free co M a ? ( os ) a ? ( sin / ) a ? sin (put ) a [ ] a ote ( ) he volume of the soli generate y the revolution of the y loi x a ( sin ) y a ( os ) a out y axis is a ( ) he volume of the soli gereate y the revolution of the y loi x a( sin ) y a ( os ) a out y axis is a ( ) ( ) in the volume of the soli generate y the revolving one ar of the y loi x a ( sin ) y a ( os ) a out x axis is a x in the volume of the soli generate y the revolution of r a os a out the initial line Sol he equation of the urve is r a os he require volume ? r / sin ? ( a os ) / sin a * os + / a , - a x in the volume of the soli generate y revolving the lemnis ate r a os a out the line Sol he given urve is r a os en e the requir volume ? r os / a ? ( os ) / os / a ? ( sin ) / os / v a ? ( sin ) / os / (Putting v sin sin v os os ) v a ? os os / v a ? os / v a ( ) a v x he ar ioi r a ( os ) revolves a out the initial line in the the volume of the soli generate Sol he given urve is r a ( os ) en e the require volume ? sin ? a ( os ) sin a * ( os ) + a ( ) a x he ar of the ar ioi r a ( os ) where / / is rotate a out the line prove that the volume generate is a Sol he given urve is r a ( os ) en e the require volume ? r sin / a ? ( os ) sin / a * ( os ) + / a , - a ote he volume of the soli forme y the revolution of the urve r a os (a )a out the initial line is a(a )Read More
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1. What is the formula for finding the volume of a solid of revolution? |
2. How do you find the surface area of a solid of revolution? |
3. Can you explain the concept of a solid of revolution? |
4. Are there any specific conditions for using the formulas for volume and surface area of solids of revolution? |
5. How can the formulas for volume and surface area of solids of revolution be applied in real-life situations? |
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