Question

Find the area under the curve y = x^2 and y = x + 1

Answer 1

Answer:

Area = $\frac{9\sqrt{5}}{4}$

Step-by-step explanation:

Given that

y = x² ......(1)

y = x + 1 ....(2)

comparing equation (1) and (2)

x² = x + 1

Rearranging

x² - x - 1 = 0 ....(3)

Solving the above equation using the quadratic equation we find the following roots

$x=\frac{1 +\sqrt{5}}{2}$ or $x=\frac{1 -\sqrt{x}}{2}$

So, by integrating the area under the curve y = x^2  from $x=\frac{1 +\sqrt{5}}{2}$ and $x=\frac{1 -\sqrt{x}}{2}$ , we can find the area under the curve y = x² and y = x + 1

$\int\limits^\frac{1 +\sqrt{5}}{2}_\frac{1 -\sqrt{5}}{2} {x^{2} } \ dx$

As we know the integration of x² is $x^{3}$. Therefore, substituting the limits

= $(\frac{1 +\sqrt{x}}{2})^{3}$ - $(\frac{1 -\sqrt{x}}{2})^{3}$

Simplifying the above expression we get the following result

Area = $\frac{9\sqrt{5}}{4}$