Integration - Areas enclosed by by y- axis?

Question

Find the area enclosed between the curve y = 1/x^2, the y-axis and the lines y=1 and y = 4 in first quadrant
please help - im struggling

can u please go through step by step - i ahve been away from school a few days and i have no idea

thank u so much - i was doing something wierd - i forgot about how 1/y^1/2 = y^-1/2
oops

Answer 1

x² = 1/y
x = (1/y)^(1/2) = y^(- 1/2)
A = ∫ x .dy between given limits
A = ∫ y^(-1/2) dy
A = 2.y^(1/2) between limits of 1 and 4
A = 2.(2 - 1)
A = 2

Answer 2

i'm getting 60 squareunits. - - - - - - - - - looking the area enclosed through... Your very last 2 applications: y = -a million and y =25 are horizontal lines decrease than and above the x-axis respectively. Your y-axis line cuts through those horizontal lines, so we are curious to no matter if the added "interesting" function y = (x^3) - 2 crosses the y-axis (esp. contained in the period -a million < = y < = 25) because if it does then you fairly'll ought to split your period of integration into 2 or perchance extra durations. So I regularly initiate with: crucial low reduce -a million to twenty-5 [integrand] dY and then if i hit upon it mandatory, i will smash the crucial into extra durations as mandatory. That performed, I observe the integrand should be a function of "y" (not "x") to be possible; so I take the function I suggested as extra "interesting" and set off with it: x = (y + 2)^ (a million/3) noticing that contained in the period -a million < = y < = 25 x continues to be perfect of the y-axis (i.e. x > 0 for the period above); for this reason i do not ought to split the period. So your integrand is (y + 2)^ (a million/3) which after integration is (3/4) [y + 2]^(4/3) to be evaluated on the limitations.

Answer 3

We can restate the question by noting that x = 1/sqrt(y)

Then, we just want to find the integral from 1 to 4 of 1/sqrt(y)

Since 1/sqrt(y) = y^(-1/2), it's integral is y^(1/2)/(1/2) = 2sqrt(y)

evaluating at y=4 and y=1, we get 2sqrt(4) - 2sqrt(1) = 4 - 2 = 2

Answer 4

easy, x = (1/y)^.5
now integrate between y values as you usually would with x.