Practice English Speaking&Listening with: Volume of solid - rt. triangle cross sections

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- FIND THE VOLUME OF THE SOLID DESCRIBED.

THE BASE OF THE SOLID IS THE DISC.

SO THAT'S DONE--THIS HERE, IT'S A CIRCLE OF RADIUS 5

ALL FILLED IN AT THE BASE.

THE CROSS SECTIONS ARE BY PLANES PERPENDICULAR

TO THE Y AXIS BETWEEN Y = -5 AND 5,

THERE ISOSCELES RIGHT TRIANGLES WITH ONE LEG IN THE DISC.

SO THIS IS REALLY WHAT THE PICTURES LOOKING LIKE.

THE BASES, AS YOU CAN SEE, ARE GOING FROM ONE SIDE

OF THE CIRCLE TO THE OTHER, PERPENDICULAR TO THE Y AXIS.

AND IT'S AN ISOSCELES SO THESE TWO LEGS ARE EQUAL,

OTHERWISE YOU WOULDN'T BE ABLE TO DO THE PROBLEM.

WOULDN'T BE DEFINED ENOUGH.

SO THESE RIGHT TRIANGLES GET BIGGER AND BIGGER AS THEY GO

TOWARD THE CENTER.

WE'RE AT THE CENTER THERE, 5, LET'S SEE IT WOULD BE,

10 ACROSS AND 10 HIGH.

SO WE'LL TAKE ONE REPRESENTATIVE CUT NOW DRAWN THIS WAY.

TRYING TO LOOK AT THE (X,Y) AXIS FROM THE TOP.

SO WE SEE THIS TRIANGULAR CUT AND WE SEE THAT THERE'S

THIS RADIUS 5 THAT COMES UP RIGHT HERE.

AND THIS IS AT OUR Y LOCATION.

THIS PART RIGHT HERE WILL BE SQUARE ROOT OF 25 - Y SQUARED.

SO THEN LOOKING OVER ON THE TRIANGLE,

THIS IS WHAT YOU HAVE.

SO THAT'S JUST THE AREA FACE OF THE TRIANGLE.

THEN THERE'S A THICKNESS GOING DOWN HERE LIKE THIS

AND THAT IS DELTA Y THICK.

SO THE VOLUME OF THE SLICE IS 1/2 BASE x HEIGHT,

THAT'S THE AREA FACE OF THE TRIANGLE,

TIMES THE THICKNESS DELTA Y.

SO PLUG EVERYTHING IN, SIMPLIFY IT DOWN TO HERE.

THE VOLUME OF THE OBJECT IS APPROXIMATELY

THE REMOND SUM OF Y VALUES GOING FROM -5 TO 5

OF THE VOLUMES SLICES.

THE VOLUME IS -5 TO 5 INTEGRAL OF 2 x 25 - Y SQUARED DY

AND YOU DO THE EVALUATION ON THAT AND IT COMES DOWN TO

1000 THIRDS CUBIC UNITS.

The Description of Volume of solid - rt. triangle cross sections