Calculus Answers

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

Suppose that f is a continuous function that satisfies

"f(x)=x\\int_{0}^{x} f(t)dt + x\u00b3"

For all x and that f(a) = 1, a ∈ R. Express f ' (a) in terms of a only:

The area under one arch of the sine curve revolves about the x-axis. Find the volume generated.

Find the surface area generated by revolving about x-axis the area in the second quadrant under the curve y=e^x.

Find the area bounded bounded by the following curves, y=x²/4 and x+4=2y.