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

Explanation:

given is a function as

`y=(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 {(27)/(x^3) } \, dx \\=(-27)/(2x^2) \\= (-27)/(2t^2)-(-27)/(2)\\=(27)/(2)(1-(1)/(t^2) )`

a) When t =10,

area =
`(27)/(2) (1-(1)/(10^2) )\\= 13.365`

b) t=100

area =
`(27)/(2) (1-(1)/(100^2) )\\= 13.49865`

c) t=10000

area =
`(27)/(2) (1-(1)/(10000^2) )\\= 13.5`