Curvilinear trapezoidLeibniz-Newton formula is helping to determine area of a curvilinear trapezoid?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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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