Question

Help with some area w/ known cross-sections problem!!! The base of a solid is the circle x2 + y2 = 4. Find the volume of the solid given that the cross sections perpendicular to the x-axis are squares.

Answer 1

The side length of each square will be 2L, which in terms of our function is 2f(x)
The area of each square as a function of x will be\[A(x)=(2L)^2=4[f(x)]^2\]the volume is the integral of area along some axis.
because the radius of the circle is 2, the integral is from x=-2 to x=2

Answer 2

as a function of x the circle is\[f(x)=\sqrt{4-x^2}\]so we can now do our integral\[\int A(x)dx=4\int_{-2}^{2}[f(x)]^2dx=4\int_{-2}^{2}4-x^2dx\]

Answer 3

we can do one last thing to make this a little easier and note that the integrand is an even function, and for all even functions we have that\[\int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\]so our integral for the volume can be written as\[V=\int A(x)dx=8\int_{0}^{2}4-x^2dx\]I think you can manage the integral yourself

Answer 4

wait can u help me with one about an ellipse

Answer 5

sure just post it separately, I will check it out

Answer 6

ok thanks