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# Definite integrals: reverse power rule

Examples of calculating definite integrals of polynomials using the fundamental theorem of calculus and the reverse power rule.

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• Why isn't "plus C" used in the anti derivatives? Is it because we are doing it for a definite integral?
• Yes this is because the integral is definite. For definite integrals, because the antiderivative must be evaluated at the endpoints and the results must be subtracted, the "plus C" terms would cancel out in the subtraction. This is why "plus C" does not appear in the answer for a definite integral.
• At I think Sal misspeaks, it is even so in the transcript. He says "So, four X to the first divided by four, well, that's just gonna be four X. " when he actually divided by one. He still did the problem correctly, but he said the wrong thing I believe.
• At , I get the algebra, but I don't understand the mathematical logic behind it. I thought that with anti-derivatives, anything with a negative derivative would cancel that same amount of anything from the positive, so to speak. e.g. my intuition would be if your lower and upper bounds of [S4dx] were (for example) -1 and 1 (or -2 and 2, etc), then the resulting triangles calculated by the definite antiderivative (4x) would cancel each other out, equalling zero, rather than netting a positive value. Tell me where I'm going wrong, please!
(1 vote)
• You have the terms negative derivative and negative antiderivative confused. When the antiderivative is negative (in other words, the area is between the x-axis and the curve below the x-axis), then it can cancel out with a positive antiderivative (area is between x-axis and curve above x-axis). However, a part of the function with a negative derivative can still have a positive antiderivative.

And, as Howard Bradley implied, areas on the left of the y-axis and areas on the right of the y-axis do not cancel out. It is areas under the x-axis and areas above the x-axis that have the potential to cancel.
• I understand how to apply the technique taught in the video. But what exactly are we finding when we take the definite integral from -3 to 5 of 4dx ? What does the value 32 tells us?

Appreciate any replies!
(1 vote)
• If we integrate the function f(x) = 4 over the interval [-3,5] and find the answer is 32, this is telling us that the area under f(x) = 4 within the interval [-3,5] is equal to 32.
• Why is f(x)'s antiderivative called F(x)? Also, who was the one who came up with calculus's format, like f'(x), d/dx, etc.?
(1 vote)
• Different mathematicians involved in the discovery and extension of calculus used different notations. For derivatives we have:

dy/dx : Leibniz notation
f'(x) : Lagrange notation
And newton notation is similar to lagrange but with a dot over the f instead.
• If you are asking why he didn't include the constant, its because it is canceled when you subtract the two evaluated antiderivatives.
• Why F(a)-F(b) gives us the area? Logically, different functions with the same a and b values will give the same area, but it is not true I think...
• I don't think -1^3 = +1. Am I right?
• You're right, but as per the video: -(-1)^3 = +1
(1 vote)
• What grade level math is this?
• 1. at , the reverse power rule was introduced with indefinite integral form, but it also applies to definite integrals, right?

2. in both examples above in the video, we don't need to consider the antiderivatives' constants?
(1 vote)
• 1. Yep! Remember that definite integrals are the same as indefinite integrals, but with an extra step of plugging in the bounds after integrating. So, any rule that applies to indefinite integrals has to apply to definite ones.

2. Definite integrals do not have a constant at the end. And there's a reason why. Here's a simple proof.

Suppose I need to find the definite integral of f(x) from a to b. So, if the antiderivative of f(x) is F(x), I get [F(x)+c] from a to b. If I plug in the bounds, I get F(b) + c - (F(a) + c). This simplifies to F(b) + c - F(a) - c. See that the constants cancel out and we get F(b) - F(a) (Also observe that this is the statement of the Fundamental Theorem of Calculus). So, even if you added a constant, it would get cancelled. So, we don't add it.