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Application of Integrals

Name: Zigya - For The Curious Learner
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Long Answer Type

21.

Draw a graph of $straight x squared over 9 plus straight y squared over 25 space equals space 1$ and evaluate area bounded by it.

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22. Using definite integrals, find the area of the ellipse

straight x squared over 4 plus straight y squared over 9 space equals space 1

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23.

Draw a graph of $straight x squared over 9 plus fraction numerator straight y squared over denominator 16 space end fraction space equals space 1$ and evaluate area bounded by it.

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Short Answer Type

24.

Using definite integrals, find the area of the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1$ .

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25.

Sketch the region of the ellipse and find its area, using integration.
$straight x squared over straight b squared plus straight y squared over straight a squared equals 1.$

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Long Answer Type

26. Find the area of the region in the first quadrant enclosed by the x-axis, the line

straight x equals square root of 3 space straight y

and the circle x² + y² = 4.

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27. Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x, and the circle x² + y² = 32.

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28.

Find the area bounded by the ellipse $straight x squared over straight a squared plus straight y squared over straight b squared equals 1 space and space the space$ ordinates x = a e and x = 0 where b² = a² (1 - e²) and e < 1.

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Short Answer Type

29. Find the area of the region bounded by the parabola y = x² + 1 and the lines y = x, x = 0 and x = 2.

We are to find the area of the region bounded by the curves.
   $straight y space equals space straight x squared plus 1 comma straight y space equals space straight x comma straight x space equals space 0$
and x = 2
Now    $straight y space equals space straight x squared plus 1$ is an upward parabola with vertex A (0, 1), y =x is straight line passing through the origin, B(2, 2) and lies below the parabola.
Now area bounded by the parabola, above the x-axis and ordinates x = 0, x = 2.
   $equals space integral subscript 0 superscript 2 left parenthesis straight x squared plus 1 right parenthesis space dx space equals space open square brackets straight x cubed over 3 plus straight x close square brackets subscript 0 superscript 2 space equals space open parentheses 8 over 3 plus 2 close parentheses space minus space left parenthesis 0 plus 0 right parenthesis space equals space 14 over 3$
Area bounded by the line y = x, above the x-axis and ordinates x = 0, x = 2.
   $equals space integral subscript 0 superscript 2 straight x space dx space equals space open square brackets straight x squared over 2 close square brackets subscript 0 superscript 2 space equals space 4 over 2 minus 0 space equals space 2$

   $therefore space required space area space equals space 14 over 3 minus 2 space equals space 8 over 3 space sq. space units.$