Curvilinear trapezoidLeibniz-Newton formula is helping to determine area of a curvilinear trapezoid?
Yes, it is helping to determine the area bounded by the limits of integration and the given curves.
For instance, to determine the area bounded by the curve f(x)=x(5+x)^4,x axis and the limits of integration x=0 to x=1, we'll apply Leibniz-Newton formula.
Int f(x)dx = F(b) - F(a), where a and b are the limit of integration.
To solve the integral, we'll change the variable:
5 + x = t
x = t - 5
We'll differentiate both sides:
dx = dt
Int f(x)dx = Int (t-5)*t^4dt
Note that it's much more easier to raise the a variable to a power then to apply Newton binomial.
Int (t-5)*t^4dt = Int t^5 dt - 5Int t^4 dt
Int (t-5)*t^4dt = t^6/6 - 5t^5/5 + C
Int (t-5)*t^4dt = t^6/6 - t^5 + C
Int f(x)dx = F(1) - F(0)
F(1) = (5 + 1)^6/6 - (5 + 1)^5
F(1) = 6^5 - 6^5
F(1) = 0
F(0) = 5^6/6 - 5^5
F(0) = 5^5*(5/6 - 1)
F(1) - F(0) = 0 - 5^5*(5/6 - 1)
F(1) - F(0) = 5^5*(1 - 5/6)
F(1) - F(0) = 5^5/6
The area of the created curvilinear trapezoid is A = 5^5/6 square units.