The region under the curves #y=x^2, y=x# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?

Question

The region under the curves #y=x^2, y=x# is rotated about a) the x axis and b) the y axis.  How do you sketch the region and find the volumes of the two solids of revolution?

Evelyn Agard · Accepted Answer

See the answer below:

Ezra Adams · Answer

undefined a) To sketch the region and find the volume of the solid obtained by rotating the region under the curves $ y = x^2 $ and $ y = x $ about the x-axis:
1. Sketch the curves $ y = x^2 $ and $ y = x $ on the same coordinate system within the desired domain.
2. Determine the points of intersection by solving $ x^2 = x $ for $ x $.
3. Set up the integral for the volume using the formula $ V = \pi \int_{a}^{b} (f(x))^2 - (g(x))^2 \, dx $, where $ f(x) $ is the upper curve and $ g(x) $ is the lower curve.
4. Integrate the expression over the interval of intersection to find the volume.

b) To find the volume of the solid obtained by rotating the region under the curves $ y = x^2 $ and $ y = x $ about the y-axis:
1. Sketch the curves $ y = x^2 $ and $ y = x $ on the same coordinate system within the desired domain.
2. Determine the points of intersection by solving $ x^2 = x $ for $ x $.
3. Set up the integral for the volume using the formula $ V = \pi \int_{c}^{d} (f(y))^2 - (g(y))^2 \, dy $, where $ f(y) $ is the right curve and $ g(y) $ is the left curve.
4. Integrate the expression over the interval of intersection to find the volume.