Find the area between the curve y=x^2 and the y axis between y=1and y=4
To find the area between the curve y=x^2 and the y-axis between y=1 and y=4, we first need to find the x-values that correspond to y=1 and y=4.
For y=1:
1 = x^2
x = ±1
For y=4:
4 = x^2
x = ±2
Now, we can set up the integral to find the area:
∫[1,4] x^2 dx = (1/3)x^3 |[1,4]
= (1/3)(4)^3 - (1/3)(1)^3
= (1/3)(64) - (1/3)(1)
= 63/3 - 1/3
= 62/3
So, the area between the curve y=x^2 and the y-axis between y=1 and y=4 is 62/3 square units.
No, you have found the area under the parabola on the x-interval [1,4]
You want the area of the horizontal strip
∫[1,4] (√y)-(-√y) dy = 2∫[1,4] √y dy = 28/3
Apologies for the mistake in my previous response. Thank you for pointing that out.
To find the correct area between the curve y=x^2 and the y-axis between y=1 and y=4, we should indeed set up the integral as the integral of the function (√y) - (-√y) with respect to y on the interval [1,4]:
∫[1,4] (√y - (-√y)) dy = ∫[1,4] 2√y dy
= 2∫[1,4] √y dy
= 2*(2/3)y^(3/2) |[1,4]
= (4/3)*4^(3/2) - (4/3)*1^(3/2)
= (4/3)*8 - (4/3)
= 32/3 - 4/3
= 28/3
Therefore, the correct area between the curve y=x^2 and the y-axis between y=1 and y=4 is 28/3 square units.