The area of the region R=(x,y):xy 27,1 y x^2 is equal to :

Question

The area of the region $R={(x,y):xy\le 27,1\le y\le {x}^{2}}$ is equal to :

Accepted Answer

$$\text{Area} = \int_1^3(x^2-1)\,dx + \int_3^{27}\left(\frac{27}{x}-1\right)dx$$

$$= \left[\frac{x^3}{3}-x\right]_1^3 + \left[27\ln x - x\right]_3^{27} = \frac{26}{3}-2+27\ln9-24$$

Since $27\ln 9 = 27\ln 3^2\cdot\ldots$ actually $\ln 27 - \ln 3$ terms combine: $27\ln27-27-27\ln3+3 = 81\ln3-27\ln3-24 = 54\ln3-24$. Adding the first bracket:

$$\text{Area} = \frac{26}{3}-2 + 54\ln3-24 = 54\ln3 - \frac{52}{3}$$

Solutions · 1