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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.
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