Evaluate the iterated integral after changing coordinate...

Transcript

00:01 This problem we want to evaluate the following triple integral we are giving the integral from 0 to 1 integral from negative square root of 1 minus x squared to the positive square root of 1 minus x squared and finally the integral from 0 to 2 minus x squared minus y squared of the square root of x squared plus y squared and the z the y the x take a look at the limits for the last two integrals this define a region the region of integration where x ranges between the values of zero and one and y ranges from the bottom function which is equal to the negative square root of one minus x squared to the positive square root of one minus x squared so if we draw a circle on the x and y plane here is my y axis there is my um x axis the region that we are referring to is the right side of um to the right side of the y axis so the region of interest is going to be this region to the right side of the y -axis notice that the radius of this disk will be one so we can express this region in terms of cylindrical coordinates that is theta will range from negative pi or two to pi or two and r will range from zero to one also in cylindrical coordinates let's write over here using cylindrical coordinates keep in mind that the v which is the z the y the x is now going to be replaced with r d z the r the theta and that x squared plus y squared is going to be replaced with r squared so this triple integral is now going to equal to the integral where theta ranges from negative pi over two to power two the integral where r ranges from zero to one and the integral where z ranges from zero to two minus r squared um here we're going to have the square of x squared plus y squared which is the square of r squared, which is just r times r d z, the r the theta.

02:04 So first, we are going to break this up into the integral from negative pi of 2 to pi over 2 of 1d theta, and then multiply this by, here the integral of r squared with respect to z is just r square z.

02:18 So we're going to have the integral from 0 to 1 of r squared times 2, so 2 r squared minus, and here we're going to have r squared times r squared, which is r to the fourth power, the r...

Question

Evaluate the iterated integral after changing coordinate systems. $$\int_{0}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \int_{0}^{2-x^{2}-y^{2}} \sqrt{x^{2}+y^{2}} d z d y d x$$

Question

Evaluate the iterated integral after changing coordinate systems. $$\int_{0}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \int_{0}^{2-x^{2}-y^{2}} \sqrt{x^{2}+y^{2}} d z d y d x$$

Please give Ace some feedback