Calculus Answers

Calculate the volume created by revolution around the axes OX, of D of the plane (OXY) where D={(x,y) ∈R: 0 ≤ y ≤x, (x-2)² +y² ≤ 4}

Simplify the sum Σ from r=0 to 2n+1 of [3/(r+1)(r+2)]

Evaluate the limit as x turns to -∞

Lim [3x + √(9x²-Ix-2I)]

Consider the following integral:

I =  ∫ 1 / (√2x²-x²) dx

(a) complete the square of f(x) = 2x²-x²

(b) use (a) together with the method of trigonometric substitution to determine the integral.

Consider the integral:

I= ∫ 5dx / 4 sin x + 3 cos x

(a) Use z - substitution to show that :

I = ∫ -10dz / (3z + 1)(z-3) where z = tan(x/2)

(b)  Now use the method of partial fractions to determine the integral.

Evaluate the integral of cos(sqrt(x))dx.

Consider the surface S = n (x, y, z) | z = p x 2 + y 2 and 1 ≤ z ≤ 3 o .(a) Sketch the surface S in R 3 . Also show its XY-projection on your sketch. (2) (b) Evaluate the area of S, using a surface integral

Let f be the function:

f(x) = ln(2x)-(2x² +3), x > 0

(a) Use the sign pattern for f'(x) to determine the intervals where f rises and where f falls.

(b) Determine the coordinates of the local extreme point(s).

(c) Find f''(x) and determine where the graph of f is concave up and where it is concave down.

(d) Find any inflection points