Calculus Answers

The area bounded by the curves y = x² + 2 and y = 3x + 2 from intersection point to intersection point is rotated around x = 3.

Determine the formula for the the solid of revolution which is obtained from this rotation.

Find the Taylor polynomial of order 4 (at x = 0) for the function : f(x) = e^{x^2}

(Sections 13.1–13.6) Let R be the region in the sketch below. MAT2615/001 (a) Describe the region R as a union of two Type 1 regions. (Use set–builder notation.) Hints: • Read the description of a Type 1 Region on p.18 of Guide 3 and study Fig. 13.9 carefully. • Shade the region R by means of vertical lines and highlight the curves which form lower and upper boundaries of R. Write the equations of these curves in the form y = g(x). • The lower boundary of R is formed by two different curves. (3) (b) Describe the region R as a union of two Type 2 regions. (Use set–builder notation.)

An curve has equation y =12/3 -2x

Find dy/dx

Consider the surface S =  (x, y, z) ∈ R 3 | z = 3 − x 2 − y 2 ; z ≥ 2 . Assume that S is oriented upward and let C be the oriented boundary of S. (a) Sketch the surface S in R 3 . Also show the oriented curve C and the XY-projection of the surface S on your sketch. (2) (b) Let F (x, y, z) = (2y, 3z, 4y). Evaluate the flux integral Z Z S (curl F) · n dS by i. determining curl F and the upward unit normal n of S and using the formula (17.2) on p. 104 of Guide 3 (5) ii. Using Stokes’ Theorem, convert the given flux integral to a line integral. 

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