The area enclosed by the loop
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 differentiation g’(x)dx =du.
From the graph, observe that the loop extends from x=-3 to x=0
And by symmetry, the area enclosed by the loop is twice the area under the top half of the curve on this interval.
The equation of top half is y=-xx+3.
Therefore, the area enclosed by the loop is
Use substitution method.
Therefore, differentiation is dx=du.
Changing the limits of integration,
At x=-3, u=-3+3=0 and
At x=0, u=0+3=3
Therefore, the integral from 0 to 3 becomes
Therefore, the area enclosed by the loop is 2453
The area enclosed by the loop is 2453