-->

Vector Algebra

Question
CBSEENMA12032703

Find the area of the region bounded by the ellipse straight x squared over 16 plus straight y squared over 9 space equals 1.

Solution
The equation of ellipse is 
              straight x squared over 16 plus straight y squared over 9 space equals 1
or        straight y squared over 9 space equals space 1 minus straight x squared over 16
or             straight y squared space equals 9 over 16 left parenthesis 16 minus straight x squared right parenthesis
therefore space space space space space space straight y space equals space 3 over 4 square root of 16 minus straight x squared end root             ...(1)
(in the first quadrant)
The ellipse is symmetrical about both the axes.
∴    required area = 4 (area AOB)
equals space 4 integral subscript 0 superscript 4 straight y space dx space equals space 4 integral subscript 0 superscript 4 3 over 4 square root of 16 minus straight x squared end root dx                       [because space of space left parenthesis 1 right parenthesis]
 equals 3 integral subscript 0 superscript 4 square root of left parenthesis 4 right parenthesis squared minus straight x squared end root dx space equals space 3 open square brackets fraction numerator straight x square root of 16 minus straight x squared end root over denominator 2 end fraction plus fraction numerator left parenthesis 4 right parenthesis squared over denominator 2 end fraction sin to the power of negative 1 end exponent open parentheses straight x over 4 close parentheses close square brackets subscript 0 superscript 4
equals space 3 open square brackets open curly brackets fraction numerator 4 square root of 16 minus 16 end root over denominator 2 end fraction plus 8 space sin to the power of negative 1 end exponent left parenthesis 1 right parenthesis close curly brackets minus open curly brackets 0 plus 8 space sin to the power of negative 1 end exponent 0 close curly brackets close square brackets
equals space 3 open square brackets open parentheses 0 plus 8 cross times straight pi over 2 close parentheses minus left parenthesis 0 plus 8 cross times 0 right parenthesis close square brackets space equals space 3 cross times 8 cross times straight pi over 2 space equals space 12 space straight pi space sq. space units. space

Some More Questions From Vector Algebra Chapter