Find the area bounded by the curve (parametric equation)?

Answerer 1

if you google desmos graphing calculator and type in the equations, you can see the space visually which helps.

x = t – (1/t) and y = t + (1/t) both have an oblique asymptote at x = y

so ultimately the area you need to find is between y = t + (1/t) and y = 65/8

we need to find where these two equation intersect.

so now we solve for t to find the point of intersection

65/8 = t + (1/t)
65/8 = (t^2+1)/t
65t = 8t^2 + 8
8t^2 – 65t + 8 = 0
(8t-1)(t-8) = 0
t = 8 or 8 = 1/8

the integral is the area under the line y=65/8 subtracting the area under the curve y=t+(1/t)

int(from 1/8 to 8) [65/8 – (t+(1/t))]dt
= int(from 1/8 to 8) [65/8 – t – (1/t)]dt

now find the anti derivative

= 65/8(t) – (1/2)t^2 – logt (now plug in 8 and solve, then plug in 1/8 and subtract that from the first)

= (65 – 32 – log8) – (65/64 – 1/128 – log1/8)
= 33 – log8 – 129/128 + log1/8