∫-22(x+3)4-x2dx by writing it as a sum of two integrals and interpret one of those integrals in terms of an area.
i) The substitution rule:
If u=g(x) is a differentiable function whose range is I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du. Here,g(x) is substituted as u and then g’(x)dx =du.
ii) Indefinite integral
∫xn dx=xn+1n+1+C (n≠-1)
iii) Integrals of the symmetric function
suppose f is continuous on -a, a
If f is odd [f(-x)=-f(x)] function, then ∫-aafxdx=0
Multiply and separate the integral term.
Consider the first integral,
Let gx=x 4-x2
Consider,g-x=-x4-x2= -x 4-x2= -g(x)
Since g is an odd function, the first integral is 0.
Now consider second integral,
Squaring both sides
This is the equation of the circle with a radius 2
The integral represents the area of the top half of a circle multiplied by 3
So, find area of semicircle with radius 2 and multiplied by 3.
Area of semicircle =12πr2
Substitute r=2 in to get 12π(2)2=2π
So required area is 3·2π=6π.