Definite integrals Evaluate the following int...

Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁴ (𝓍 ― 2)/√𝓍 d𝓍

Accepted Answer

Welcome back, everyone. In this problem, we want to compute the integral of X minus 8 divided by root X with respect to X between the limits of 4 and 16 using the fundamental theorem of calculus part 2. A says it's 16/3, B negative 163, C 8/3, and a D negative 83. Now, before we can use the fundamental theorem of calculus to compute our integral, let's first evaluate the integral of the expression, OK? So evaluating. All right, in you go. Let me just write a note here. OK. Then that means That we are finding the integral of X minus a divided by root X with respect to X. Now we can simplify by splitting up our terms, so that's really the integral of X divided by root X with respect to X minus the integral of a divided by root X with respect to X. Now if we use what we know about powers, then really in our first we're finding the integral of X to the power of a half with respect to X, because this, in this case, we would subtract the power of a half from 1 in our powers for X. And in the second term, if we take out our constant it, then we're finding the integral of x to the pore of a negative half with respect to X. Now we can integrate both of these, or we can find the antiderivative by using the power. Because in our first term, the integral of X to the half of D X, if we think about it, means that we're going to add 1 to the power, so it's gonna become X to the power of three halves, and we're going to divide by the new power. So that's gonna give us 2/3, multiplying x to the power of 3 halves, plus some constant C1. On the other hand, we're subtracting it multiplied by our integral of X to the power of negative the X and here if we add 1 to the power, that's going to be X to the power of half, we divide by our new power a half that would be 2. So this is 2 X1/2 plus a second constant C2. And know if we combine both our constant C1 and C2 into some other constant C, then this integral becomes 2/3 of X to the power of 3 halves minus 16 multiplied by X to the 1/2 plus some constant c. Know that we have the integral of our expression, OK, then we can compute the integral using the fundamental theorem of calculus, part two. So what does that theorem say? Well, if we recall the theorem, OK. The theorem basically tells us that if we are finding the integral of a function F of X with respect to X between the limits A and B, then it's actually equal to the difference between the antiderivatives evaluated at B and A. So, since we already know our limits, our lower limit is 4, let me just put that here. Our lower limit is 4, and we know that our upper limit is 16. Then we can evaluate these for our antiderivative and then find the difference between them, OK? So let's go ahead and do that to compute our integral. Now, starting with our upper limit, OK, so we're substituting 16 into our anti derivative. That's going to become 2/3 of 16 rates to the power of three halves minus 16 multiplied by 16 to 2, OK? And now if we think about it here. 16 rates to the report of 3 halves is 64. But 16 race to a half is 4, cuz that's really the root of 16. I know this is going to be 2/3 of 64 minus 64. And again, if we think about that, that's really going to be -1/3 of 64. Or we can write that as -64/3. On the other hand, If we evaluate the antiderivative again at X equals 4, then that's gonna be 2/3 of 4 to the power of 3 halves, minus 16 multiplied by 4 to the power of 2, and that's really 2/3 of it. Minus 16 multiplied by 2, which if we evaluate this again and leave it as thirds is going to be -803. So now that we have evaluated our antiderivative at the lower and upper limit, then we can subtract because no, that means our result. OK, is going to be. Equal to the difference between the value at 16-64/3. And the value at 4, that is negative 803. So that's -64/3 minus -803, and that's gonna be -64/3 plus 80/3 which would be equal to 16/3. Therefore, the integral of X minus 8 divided by root X with respect to X between the limits 4 and 16 is 16/3. If we look back at our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁴ (𝓍 ― 2)/√𝓍 d𝓍

Key Concepts

Definite Integrals

Fundamental Theorem of Calculus

Antiderivatives

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