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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

148 Views

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.

112 Views

Short Answer Type

24.

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

259 Views

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.

637 Views

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.

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Long Answer Type

30. Find the area of the region bounded by the curves y = x² + 2, y = x, x = 0 and x = 3.

We are to find the area of the region bounded by the curves
y = x² + 2,
y = x,
x = 0
and x = 3
Now y = x² + 2 is an upward parabola with vertex (0, 2).
y = x is a straight line passing through the origin,
(3, 3) and lies below the parabola.

Now area bounded by the parabola, above the x-axis and ordinates x = 0, x = 3
$equals space integral subscript 0 superscript 3 left parenthesis straight x squared plus 2 right parenthesis space dx space equals space open square brackets straight x cubed over 3 plus 2 straight x close square brackets subscript 0 superscript 3 space equals space open parentheses 27 over 3 plus 6 close parentheses space minus space left parenthesis 0 plus 0 right parenthesis space equals space 15$
Area bounded by the line y = x, above the x-axis and ordinates x = 0, x = 3
$equals space integral subscript 0 superscript 3 straight x space dx space equals space open square brackets straight x squared over 2 close square brackets subscript 0 superscript 3 space equals space 9 over 2 minus 0 space equals space 9 over 2$
$therefore space space required space area space space equals space 15 space minus space 9 over 2 space equals space 21 over 2 space sq. space units. space$