Find the value of the integral by using an area formula from geometry?
1) the integral from 1/2 to 1 for sqrt(1- x^2)
2) The integral from 0 to 1 for sqrt (2 – x^2)
I know that the graph of these are circles and to use πr^2 but I don’t know how…? Thanks
Your teacher is being unnecessarily cruel by making you use the formula for the area of a circle to evaluate the integral. If you could argue that these are pie slices, it would be much easier, but they aren’t.
Look, the indefinate integral relative to x of sqrt(r^2 – x^2) is…
∫sqrt(r^2 – x^2) dx = 1/2*x^2*sqrt(r^2 – x^2) + 1/2*r^2*arctan(x/sqrt(r^2-x^2)) + C
Call this I(x) = 1/2*x^2*sqrt(r^2 – x^2) + 1/2*r^2*arctan(x/sqrt(r^2-x^2)) + C
Your arctangent function will return in radians, so have your mode set accordingly.
For part 1: r = 1. Plug in x=1 and x=1/2 to the indefinite integral result. Subtract I(1) – I(1/2) and you got your result. We do need to replace the arctangent function with Pi/2, because it is the arctangent of an expression containing division by zero. 0.8307 units^2 is the answer. The analytical answer is a mess, so no need to show.
For part 2: r = sqrt(2). Plug in x=1 and x=0 to the indefinite integral result. Subtract I(1) – I(0) and you got your result. 1/2 + Pi/4 gives you the answer, or 1.285 units^2.
Your teacher is being unnecessarily cruel by making you use the formula for the area of a circle to evaluate the integral. If you could argue that these are pie slices, it would be much easier, but they aren’t.
Look, the indefinate integral relative to x of sqrt(r^2 – x^2) is…
∫sqrt(r^2 – x^2) dx = 1/2*x^2*sqrt(r^2 – x^2) + 1/2*r^2*arctan(x/sqrt(r^2-x^2)) + C
Call this I(x) = 1/2*x^2*sqrt(r^2 – x^2) + 1/2*r^2*arctan(x/sqrt(r^2-x^2)) + C
Your arctangent function will return in radians, so have your mode set accordingly.
For part 1: r = 1. Plug in x=1 and x=1/2 to the indefinite integral result. Subtract I(1) – I(1/2) and you got your result. We do need to replace the arctangent function with Pi/2, because it is the arctangent of an expression containing division by zero. 0.8307 units^2 is the answer. The analytical answer is a mess, so no need to show.
For part 2: r = sqrt(2). Plug in x=1 and x=0 to the indefinite integral result. Subtract I(1) – I(0) and you got your result. 1/2 + Pi/4 gives you the answer, or 1.285 units^2.
References :
You should use formula for calculating area of circular segment
http://en.wikipedia.org/wiki/Circular_segment
1) Integral gives area equal to half area of the circular segment for x from 1/2 to 1 of the unit circle.
A = (1/2) ·(R²/2)·(θ – sin(θ))
R = 1, θ = 120° = 2π/3
A = (1/4) (2π/3 – sin(2π/3)) = (2π/3 – √3/2)/4 ≈ 0.307
If you have doubts just give me a message. Good luck!
References :
http://en.wikipedia.org/wiki/Circular_segment