Calculus

R is the region in the plane bounded below by the curve y=x^2 and above by the line y=1.

(a) Set up and evaluate an integral that gives the area of R.

(b) A solid has base R and the cross-sections of the solid perpendicular to the y-axis are squares. Find the volume of the solid.

(c) A solid has base R and the cross-sections of the solid perpendicular to the y-axis are equilateral triangles. Find the volume of the solid.

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  1. it is symmetric around the y axis so do the integral from x = 0 to x = 1 and double it
    2 int from 0 to 1 of x^2 dx
    2 (x^3)/3 = 2/3

    length of side of square = 2x^2
    area of cross section = 4 x^4
    integrate 4 x^4 dx from 0 to 1

    now do the same for area of cross section = x^4 sqrt 3

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  2. I did the area below the line. For between use (1-x^2) dx etc

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  3. What is symmetric around the y-axis? This is confusing because I don't really know how to draw the solid. First it says the solid has the base of R. So does that mean R should be on the x-axis? But how do you find out the length of the side of the square? Is one side of the square solid bounded by the region so that one of the sides of the square faces of the solid supposed to have a length that is from one intersection point of both curves to the other intersection point? Same goes for part c. Thanks

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