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The second part of the Fundamental Theorem of Calculus and the remarkable relationship between integration and differentiation.

The Fundamental Theorem of Calculus has far-reaching applications, making sense of reality from physics to finance. The second part gives us a way to compute integrals.

When downloading a file, the number of bytes downloaded can be found by integrating the function describing the download speed as a function of time using the second part of the Fundamental Theorem of Calculus.
To find how much money a product or service brought in during a particular time frame, a company can integrate the marginal revenue over that time. The second part of the Fundamental Theorem of Calculus allows us to perform this integration.
Physicists use integration — made possible by the second part of the Fundamental Theorem of Calculus — to measure a variety of quantities, such as energy, work, inertia, and electric flux.

FUNDAMENTAL THEOREM OF CALCULUS - 2
Fb-Fa=∫abfxdx
Fx=2x-2tan-1⁡x,   -3≤x≤4
fx=x2-1x2+1+1,  -3≤x≤4
Fx=sin⁡x,   -5π2≤x≤5π2
fx=cos⁡x,   -5π2≤x≤5π2
Fx=2x-7,0≤x≤4148x3+x-133,x≥4
fx=2,0≤x≤4116x2+1,x≥4
Ft=∫0tfx dx      =∫0t2x-1.53 -2x+7dx       ≈Approx value
Ft=∫-3π2tfx dx       =∫-3π2txsin⁡x-π3+5dx        ≈Approx value
Ft=∫0tfx dx       =∫0t25x2-1 dx        ≈Approx Value
Drag the sliders to adjust the values of a and b. The vertical red arrow in the left graph represents the distance between F(a) and F(b).
1 2 3
1. When is ∫ baf(x) dx negative?
2. If a < b, then what can we say about F(b) - F(a)?
3. For which of the following choices of a and b is the value of
∫ baf(x) dx largest
Drag the sliders to adjust the values of a and b. The vertical red arrow in the left graph represents the distance between F(a) and F(b).
1 2 3
1. When is F(b) - F(a) positive? Check all that apply.
2. For which of the following choices of a and b is
∫ baf(x) dx = 0? Check all that apply.
3. For which of the following choices of a and b is
∫ baf(x) dx = -1? Check all that apply.
Drag the sliders to adjust the values of a and b. The vertical red arrow in the left graph represents the distance between F(a) and F(b).
1 2 3
1. When the red arrow in the left graph is pointing down, what is true? Check all that apply.
2. Which of the following statements is true?
3. Over which of the following intervals does F have the greatest net increase (that is, when is
F(b) - F(a) greatest)?
Result