Question

Find the area of the region bounded by the hyperbola 25x2 − 4y2 = 100 and the line x = 3. (Using trigonometric substitution)

Answer 1

23.92

Explanation:

We solve for y:

`25x^2-4y^2=100`

`y=5√(x^4/4-1)`

use trig substitution:

`x=2secu`

`x^2=4sec^2u`

The derivative of x is:

`dx=2secutanu du`

when x=2
`u=0`

when x=3
`u=sec^-^1\cdot(3/2)`

The area is defined as 2xarea:

The area is the integral of the equation:

`A=2\int\limits^a_b {5√(x^2/4-1) } \, dx` for range 0 to sec-1(3/2)

Substitute x=2secu

`A=\int\limits^a_b 10{√((4/4)sec^2u-1) } \, dx`

`A=10\int\limits^a_b {√(sec^2i-1)23secutanu } \, du`

We know that sec²-1 = tan²u

`A=20\int\limits^a_b {tan^2u secu} \, du`

`A=20\int\limits^a_b {(sec^2u-1)secu} \, du`

`A=20[\int\limits^a_b {sec^3u} \, du - \int\limits^a_b {secu} \, du]`

After simplifying

`A=10[secutanu-ln(secu+tanu)]`

For the range

`A=10[(3/2)√(5) /2-ln(3/2+√(5)/2)]`

`A=23.92`