fundamental theorem of calculus

Question

Use the Fundamental Theorem of Calculus to find the "area under curve" of  between  and .

Answer 1

The Fundamental theorem of calculus states that if a function f is continuous on the interval [a, b] the:

the INTEGRAL f(x) from a to b of the function is f(x) is F(b) -F(a) where the derivative of F, indicated by F' is f(x) over he interval.

In simple words: 1) find the integral of the given function f(x)

2) Calculate the value of the integrated function at point b- indicated as F(b) -

and at point a - indicated as F(a).

3) Substract F(a) from F(b) which is simply F(b) - F(a).

In our case F - the integral of f(x) = 4x + 6 is simply F(x) = 2(x)2 + 6x.

At point b = 3 the value of F(3) = 2x9 + 18 = 36 and at point x=1 the value of F(1) = 10.

Subtraction F(3) - F(1) = 28.

You can verify your calculation by noting that the surface under the funcntion f(x) = 4x +6 from x=1 to x=3 is a trapeze with a height of 3-1=2 and the bases are f(1) =10 and f(3) = 18 respectively.

The area of the trapeze is A= {((b1 +b2) xh }/2 which is{( 10+18)x2}/2 = 28

Answer 2

Area A = ∫₁³(4x + 6)dx = (2x² + 6x)₁³ = (2·3² + 6·3) - (2·1² + 6·1) = 28