Question

Find the perimeter of the region bounded by y=x and y=x2.
you are to set up an integral that represents the volume when the region is rotated.

Answer 1

Quite a long problem! To find the perimeter of the region bound by the two curves, you must find the length of the curves and add them up. The equations intersect at (0,0) and (1,1). Finding the length of y = x from x = 0 to x = 1 is easy; you can just use the distance formula, or the Pythagorean theorem.\[d = \sqrt{\Delta ^{2}x + \Delta ^{2}y} = \sqrt{(x _{1}^{}-x _{2}^{})^{2}+(y _{1}^{}-y _{2}^{})^{2}}\] You can solve for that and you should get sqrt (2). Finding the length of the curve y = x^2 is unfortunately much harder. The equation for the length of this curve from 0 to 1 should be...\[L = \int\limits_{a}^{b} \sqrt{1 + (\frac {dy}{dx}^{})^{2}} dx = \int\limits_{0}^{1} \sqrt{1 + (2x^{})^{2}}dx\] Adding the two distances should give you the perimeter of the region.
Finding the volume is much easier. I'm assuming that the region is rotated about the x-axis.\[V = \pi \int\limits_{0}^{1} (x)^{2} - (x^2)^{2} dx\] I hope this helped, good luck!