If A is the area in the first quadrant enclosed by the curve C: 2 x...

Given,

$y = 2{x}^{2} + 1$

Now tangent to above equation at $\left(1, 3\right)$ is given by,

$y - 3 = 4 \left(x - 1\right)</p><p>$⇒ y = 4x - 1

Now plotting the $diagram we get,<br /><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/53322922-5fbf-4b5c-a785-a38019933fae.png" style="width: 350px; height: 329px;" /><br />Required area of bounded region will be,</p><p>A = {∫}_{0}^{1}\left(2{x}^{2} + 1\right)dx - \text{area of} △ QOT - \text{area of} △ PQR + \text{area of} △ QRS$

$⇒ A = \left(\dfrac{2}{3} + 1\right) - \dfrac{1}{2} - \dfrac{9}{8} + \dfrac{9}{40} = \dfrac{16}{40}$

$⇒ 60A = 16$