Question

Find the area under the curve y = 27/x3 from x = 1 to x = t. Evaluate the area under this curve for t = 10, t = 100, and t = 1000. Find the total area under this curve for x ≥ 1.\

Answer 1

Answer:

Step-by-step explanation:

given is a function as

$y=\frac{27}{x^3}$

We are to find the area form x=1 to x=t

The curve from x=1 lies in the I quadratnt.

So area  above x axis is to be calculated

Area = $\int\limits^t_1 {\frac{27}{x^3} } \, dx \\=\frac{-27}{2x^2} \\= \frac{-27}{2t^2}-\frac{-27}{2}\\=\frac{27}{2}(1-\frac{1}{t^2} )$

a) When t =10,

area = $\frac{27}{2} (1-\frac{1}{10^2} )\\= 13.365$

b) t=100

area = $\frac{27}{2} (1-\frac{1}{100^2} )\\= 13.49865$

c) t=10000

area = $\frac{27}{2} (1-\frac{1}{10000^2} )\\= 13.5$