Question 1 Calculus

Question

A plane region is bounded by the curve y  =  x² ,  y  =  4x ,  x  =  0  and x  =  1.



a)         Find the area of the plane.

b)         Calculate the centroid of the plane.    

Answer 1

You're not looking for the area or the centroid of the plane: you're wanting those things for the smaller region inside the plane described in the problem.

Are you in Calc 2 or Calc 3? Graph the curves. You'll notice that the line y=4x lies above the parabola y=x^2 when you consider x between 0 and 1. Therefore, the area will be ∫₀¹(4x-x^2)dx, or equivalently ∫₀¹∫_x^2^4x dydx (only use this second choice if you're in Calculus 3).

As for the centroid, if you're in Calculus 2, to find the y-coordinate, you have to re interpret the area of your region as a sum of dy integrals (i.e. you identify "left x" and "right x" that apply to a range of y-values. In this case, for y between 0 and 1, x=(1/4)y (that is, y=4x rewritten) is on the left and x=√(y) (that is, y=x^2 for x≥0) is on the right; however, for y between 1 and 4, we have x=(1/4)y on the left and x=1 on the right. So, our area may be considered as ∫₀¹{[√(y)]-(y/4)}dy+∫₁⁴{1-(y/4)}dy - if you're in Calculus 3, in double integrals this is ∫₀¹∫_y/4^√ydxdy +∫₁⁴∫_y/4¹dxdy. Now you're ready to find the centroid: in short, take the integrand of the x integral and multiply it by x, then evaluate the resulting integral to get the x-coordinate of the centroid; for the y-coordinate of the centroid, take the two y-integrals, multiply their integrands by y, evaluate the two resulting integrals and add the two resulting numbers together to get your desired y-coordinate.

The Calculus 3 method for this problem is significantly easier: the x-coordinate of the centroid is ∫₀¹∫_x^2^4x xdydx, and the y-coordinate of the centroid is ∫₀¹∫_x^2^4x ydydx.

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