5

Q5.Evaluate the triple integral J0 J (yz zx + xy)dxdydz05 Marks CLO (51...

Question

Q5.Evaluate the triple integral J0 J (yz zx + xy)dxdydz05 Marks CLO (51

Q5. Evaluate the triple integral J0 J (yz zx + xy)dxdydz 05 Marks CLO (51



Answers

Use (5) and the summation formulas in Theorem 5.3 .2 to evaluate the given definite integral. $$ \int_{0}^{2}\left(3-x^{3}\right) d x $$

Were asked to evaluate this integral here. Now upon applying our general power rule for intros, we find this is equal to one half X squared, Evaluated at 1 -3. It's now we can plug in to evaluate so we're left with one half times the quantity, one squared -30 sq. If you work this out a little bit, use this with one half times the quantity one minus nine. And if we take 1 -10 who have put negative eight and one half negative eight B negative four, and that is our final answer.

So for this integral we're going to use you do your substitution and we're just gonna pick a. You who's um derivative D. U. Is also part of this function. And here the denominator eat the Z plus. See if we took the derivative of that, it would be equal to the numerator, E. To the Z plus one. Since the derivative of E. To the Z. Is itself. And the derivative of just see is one. And so we're gonna let you hear equal E. To the Z plus C. So then D. U. Is equal to eat the Z plus one and then multiplied by D. C. And so we're just going to make these two substitution into our integral. And so we now are looking at the integral from 0 to 1 of um D. You divided by. Yeah. And so this integral is pretty straightforward. Um One divided by you. The integral of that. It's going to be the anti derivative of one divided by you. Which is the natural log of you. Since the derivative of the natural log of U. Is equal to one divided by you. So we're looking at the natural log of you From 0 to 1. And what we're going to do is we're gonna plug in what U. Is equal to. So it's E. To the Z power plus C. This is equal to the natural log of E. To the sea plus C. From 0 to 1. So N. Z. Is equal to one. We get E plus one. The natural log of E plus one. And then we have minus when Z. is equal to zero. We have the natural log E to the zeroth power, which is one plus zero. So we're gonna have the natural log of one and the natural log of one is equal to zero, so this is just equal to the natural log of E plus one.

Were asked to evaluate it broke here Now, upon applying our general power rule for integral, those we find this integral is equal to 1/2 x squared. This would be evaluated at 30 stomach plug into a price, just be equal to 1/2 times the quantity three squared minus zero squared. If we work this out to one half of nine, And this is just equal to 9/2, and that is our final answer.

If we take you to be five to the three x plus two than D u D E X. That's equal to three times Ellen of five times five to the three x by chain room. And now we can replace five to the three x dx by do you over three times the natural log of five. So this becomes 1/3 times Ellen of five times the inter groom of One Over You, which is equal to 1/3. Ellen of five times Ellen of five to the three x plus two plus C. And we don't have to worry about the absolute values because five to the three X is positive. It's greater than zero, and that completes the problem.


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