Calculus Answers

find the centroid of the region with the indicated boundaries y = 4 - x^2 and the x-axis

find the length of the arc of the curve 9y²= (x²+2)³ from the point where x=0 To the point where x = 2.

If f(x,y) = x^3 + 4xy^2 + y^3, show that f(ax, ay)= a^3 f(x,y).

If f(x) =10^x and Φ(x) =log10 x, show that f[Φ(x)] = Φ [f(x)] = x.

If f(x)= x^2 -1 and g (x) =2x +1, show that f[g(x)]=4x(x+1).

If P(x) =√x, show that P(x+h) - P(x) = h(√x+h + √x).

If Φ (r)= 2^r, show that Φ (r+1) =2 Φ(r).

If F(z)= log z, show that F(xy) = F(x) +F(y).

Using the methods suggested in the preceding problem, find the area of the trapezoid bounded by the line y= x+3, the ordinates x=1, x=3, and the x axis.