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Application of Integrals

Name: Zigya - For The Curious Learner
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Long Answer Type

11. Find the area of the region bounded by the curve

straight y space equals space 2 square root of 1 minus straight x squared end root

and x-axis. Draw a rough sketch..

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Short Answer Type

12. Calculate the area bounded by the parabola y² = 4 a x and its latus rectum.

The equation of parabola is y² = 4 a x. ...(1)
Let O be the vertex, S be the focus and LL' be the latus rectum of parabola.
The equation of latus rectum is x = a.
Also, we know that parabola is symmetric about x-axis.

therefore space space space required space area space space equals space 2 space left parenthesis area space OSL right parenthesis

equals space 2 integral subscript 0 superscript straight a straight y space dx space equals space 2 space integral subscript 0 superscript straight a 2 square root of straight a space square root of straight x space dx equals space 2 space.2 square root of straight a integral subscript 0 superscript straight a straight x to the power of 1 half end exponent dx space equals space 4 square root of straight a open square brackets fraction numerator straight x to the power of begin display style 3 over 2 end style end exponent over denominator begin display style 3 over 2 end style end fraction close square brackets subscript 0 superscript straight a equals space 4 square root of straight a.2 over 3 open square brackets straight x to the power of 3 over 2 end exponent close square brackets subscript 0 superscript straight a space equals space fraction numerator 8 square root of straight a over denominator 3 end fraction space open square brackets straight a to the power of 3 over 2 end exponent minus 0 close square brackets equals space fraction numerator 8 square root of straight a over denominator 3 end fraction. straight a to the power of 3 over 2 end exponent space equals space 8 over 3 straight a squared space sq. space units.

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13.

Give the rough sketch of the curve y² = x and the line x = 4 and find the area between the curve and the line.

150 Views

14.

Find the area bounded between the curve y² = x and the line x = 3. Draw the rough sketch also.

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15. The area between x = y² and x = 4 is divided into two equal parts by the line x = a, find the value of a.

741 Views

16.

Find the area of the region bounded by the curve y = x² and the line y = 4.

220 Views

17.

Find the area of the region bounded by the curve y² = 4x and the line x = 3.

341 Views

18.

Using definite integrals, find the area of the circle x² + y² = a².

287 Views

Long Answer Type

19. Find the area of the region bounded by the ellipse

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

136 Views

Short Answer Type

20. Find the area of the region bounded by the ellipse

straight x squared over 16 plus straight y squared over 9 space equals 1.

151 Views

Previous Year Papers

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Test Yourself

Long Answer Type

Short Answer Type

12. Calculate the area bounded by the parabola y2 = 4 a x and its latus rectum.

Long Answer Type

Short Answer Type

12. Calculate the area bounded by the parabola y² = 4 a x and its latus rectum.