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  4. The figure shows as triangle AOB and the parabola y = x2. The ratio of the area of the triangle AOB to the area of the region AOB of the parabola y = x2 is equal to

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Q. The figure shows as triangle AOB and the parabola $y = x^2$. The ratio of the area of the triangle AOB to the area of the region AOB of the parabola $y = x^2$ is equal toMathematics Question Image

Application of Integrals

Solution:

Area of $\Delta AOB = \frac{1}{2} \times$ base $\times$ height
$= \frac{1}{2}\times 2a\times a^{2} = a^{3}$ units
Area of region AOB
image
$= 2\,\int\limits^{a^2}_{0} x\, dy = \int\limits^{a^2}_{0} \sqrt{y} \,dy$
$= 2 \left[\frac{y^{3/ 2}}{3/ 2}\right]^{a^2}_{0} = \frac{4}{3} a^{3}$ units
$\therefore $ ratio of areas $= \frac{a^{3}}{\frac{4}{3}a^{3}} = \frac{3}{4}$