Green’s Theorem, Triangle, x^2y^2+4xy^3
Transcripts
Hello everyone, Tom from everystepcalculus.com and everystepphysics.com. You can buy my programs to pass your physics and calculus classes step by step on all the problems which you need to show your work. Index 8 to get to my menu. We’re gonna do the Green’s Theorem, today, in this one. And they give us a triangle with vertices to figure out the double integral. So we’re going to choose index 8 to get to my menu and we’re going to scroll down here to Green’s Theorem it’s alphabetical. There’s Green’s theorem there. They give us the x or the i function from the line integral. You have to press Alpha before you enter anything in these entry here. Alpha x squared times y squared. And then we’re going to choose for the y the J function. Alpha 4 times x times y cubed. I always show you what you’ve entered. Here is the line integral, it really should really be called the curved integral. There are no integrals on lines. Straight things, everything needs in exponent to work. That’s the reason they use cosine and sine so much because it’s a sine wave which is a perfectly smooth curve and the use balls and spheres and etc, to do all their calculations, calculations as I see it. Calculus solves nothing in real life does nothing for anybody. Just a Soduko of math. Something that people do to waste time just like crossword puzzles in English. So anyways there’s a line integral of it and I’m going to say it’s ok to go with. P and q system or m and n system. I use the the n and m, depending on which professor you’re dealing with or the book. Compute the partials, partial of m with a partial of y is this. Partial of n with a partial of x divided by the partial of x is this . And we’re going to subtract the m from the n partials. And this right here. Ok and so now, we’re going the triangle, which they give us. We are going to choose a number to the points. They don’t give us y y x, they give us 3 points. First point is 0 0 , second point is 1 and 3, we’re going to choose number 3, scroll to it. We’re gonna add it. Alpha 1 alpha 3. And the next point is 0, 3 so we’re going to use 0 and then the question mark here, number 1
we’re going to enter number 3, alpha 3. And again, we look and see if we have entered correctly, and we have. Enter umber 1 or scroll to it. We’re going to set up the Green’s Theorem. Double integral 01 in three x Notice this, if you do the vertices and put it on a piece of paper notice that the line here is an angle such as this where certain parameters and this is the y function remember the y equals mx+b that you learned in Algebra? Well the mx is the slope and of course 3 over 1 is the slope of this line. So if you graphed it, you’d put 3x into the Y function on your graphing calculator and that would be the angle just like this. And then of course we have 3 over 0 and then 3 down the origin of the graph. So that’s the triangle we’re working with. And we’re going to integrate this right here with respect to Y and over this range here. And that equals this over the range. We take the 3x and put it in for all the y’s and we come up with 72 x to the 4 and then 0 of course is a 0. And upper minus lower is 72 x to the fourth. Now we’re going to integrate that with respect to x over this range here which equals 72x to the 5. Remember we add 1 to the exponent and divide by 5. That exponent in the problem. And over this range here. Remember Newton and Leibniz were working on exponents in the sixteen hundreds, that’s how they came up with calculus and of course logarithms, also. Logarithms is an exponent. And so we’re gonna do that range at X equals one. we substitute that for the x, come up with this, and the answer is 72 over 5 square units. 14.4 square units. Pretty neat, huh? everystepcalculus.com. go to my site buy my programs. It’s the best deal around. Much, much more economical and worth worth it then they even the calculus book. So enjoy my program and pass calculus or physics. Have a good one!
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