Integrate Multiple Number with airSlate SignNow

Integrate multiple calculated

in this video we're going to focus on double integrals so how can we evaluate the iterated integral that we see on the board here before we begin the first thing we need to realize is that the numbers 1 and 3 correspond to y those are y values and notice that dx is written later therefore 0 and 2 are x values so just keep that in mind what we're going to do is we're going to work from the inside towards the outside so we're going to evaluate this integral in the middle so let's do that over here so we have the integral from 1 to 3 x y squared d y so d y tells us that we're integrating with respect to y which means that we're going to treat x as if it's a constant so what is the antiderivative of y squared the antiderivative of y squared is going to be y to the third divided by 3. you need to add one to the exponent and then divide by the result you get so now we're going to evaluate it but since we have two variables and x and the y we're going to specify that we're replacing y with one and y with three so this is gonna be x over three and then we're gonna replace y to the third with three so it's gonna be three to the third minus one to the third three to the third power is 27 one to the third is one so this gives us 26 over three times x so we're going to replace what we have in the brackets with what we see here so right now we have the integral from 0 to 2 of 26 over 3x dx so now let's find the antiderivative of x to the first power so we're going to add 1 to the exponent it's going to become 2 and then divide by that result now we're going to plug in 0 and 2. we don't need to write x equals 0 and x equals 2 because we only have one variable in that expression 26 divided by 2 is 13. so we have 13 over 3 times x squared and we're going to evaluate from 0 to 2. so let's take out the constant and let's replace x with 2 so that's going to become 2 squared and then once we replace with 0 we have that 2 squared is 4 and 13 times 4 is 52 so the final answer for this problem is 52 over 3. in the last problem we saw that the value of this double integral was equal to 52 over 3. but what's going to happen if we change the order of integration let's say if we put the one to three first what's going to happen it turns out that the value of these two integrals are the same now keep in mind the numbers one and...