The area of the region bounded by the curves x = at^2 and y = 2at between the lines parallel to y-axis corresponding to t = 1 and t = 2 is equal to : Option 1: 14/3a^2 Sq units Option 2: 112a^2/3 Sq units Option 3: 56a^2/3 Sq units Option 4: 28/3a^2 Sq units

Answer: 56a^2/3 Sq units. Solution: Eliminating t gives the parabola y^2 = 4ax, and t=1, t=2 give the ordinates x=a and x=4a. Using symmetry about the x-axis, area = 2_a^4a 2√ax dx = 8√a/3(8a^(3/2)-a^(3/2)) = 56a^2/3. Note: The two given equations describe one parabola; we have taken the region bounded by the whole curve between the two ordinates.

Calculus

Application of Integrals

Medium

core

The area of the region bounded by the curves $x = at^2$ and $y = 2at$ between the lines parallel to $y-axis$ corresponding to $t = 1$ and $t = 2$ is equal to :

$\dfrac{14}{3}a^2$ Sq units

$\dfrac{112a^2}{3}$ Sq units

$\dfrac{56a^2}{3}$ Sq units

$\dfrac{28}{3}a^2$ Sq units

✅ Correct Option: 3

Eliminating $t$ gives the parabola $y^2 = 4ax$ , and $t=1$ , $t=2$ give the ordinates $x=a$ and $x=4a$ . Using symmetry about the $x$ -axis, area $= 2\int_a^{4a} 2\sqrt{ax}\,dx = \frac{8\sqrt{a}}{3}\left(8a^{3/2}-a^{3/2}\right) = \frac{56a^2}{3}$ .

Note: The two given equations describe one parabola; we have taken the region bounded by the whole curve between the two ordinates.

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The area of the region bounded by the curves x=at2x = at^2x=at2 and y=2aty = 2aty=2at between the lines parallel to y−axisy-axisy−axis corresponding to t=1t = 1t=1 and t=2t = 2t=2 is equal to :

Solution

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The area of the region bounded by the curves $x = at^2$ and $y = 2at$ between the lines parallel to $y-axis$ corresponding to $t = 1$ and $t = 2$ is equal to :