Find if they exist points at which the tangents are

Question

1. Evaluate or determine the definite integral

(a) ₁∫^N(1/t⁴) dt

(b) _h∫²e^-t dt

(c) ∫xe^{-x^2} dt

(d) _-3∫²cos πt dt

2. Find the volume of the solid obtained by rotating the triangle (6,0)(6,8)(2,2) about the horizontal axis. Clearly include a sketch and set up the integral first.

3. Let C be the curve y = 5x^1.5 where 1 ≤ x ≤ 9. Calculate the length of C.

4. (a) Tabulate a few points of the curve: x = 0.5t², y = 0.5 (t - 1)².

(b) Hand sketch a possible graph.

(c) Find, if they exist, points at which the tangents are horizontal.

(d) Find, if they exist, points at which the tangents are vertical.

5. Show that if b ≠ 0.