Consider the region bounded by the curve y 2x2 - x3 and

Question

1. Choose TWO of the integrals below and evaluate on the next page(s).

1. ∫ln(x) dx

2. ∫1/(1 + 4x²)dx

3. ∫cos(x)e^5xdx

4. ∫cos² x dx

5. ∫6y ^1.5 + sec²(y) + 7 dy

6. ∫cos²(x) sin³(x) dx

7. ∫te^-tdt

8. ∫x/√(1 - x²)dx

9. ∫x sec² x dx

10. ∫x/(x + 23)dx

2. Consider the region bounded by the curve y = 2x² - x³ and x-axis on [0, 2]. Determine the volume of the solid obtained when this region is rotated about the y-axis.

3. Assume that the formula

∫(ln x)ⁿ dx = x (ln x)ⁿ - n∫(ln x)^n-1dx (1)

is true for all positive integers n.

(a) Use equation (1) to compute ∫_o¹(ln x)²dx.

HINT: This means set n = 2.

(b) Use equation (1) to compute

∫_o¹(ln x)³dx.

HINT: This means set n = 3.

(c) Make a conjecture on the value of ∫_o¹(ln x)ⁿdx.

4. You have just accepted a position with the Department of Defense working on ballistics. Your first assignment is as follows: Consider the region R bounded by the curve y = √(4 - x²) for 0 ≤ x ≤ 1, the line y = 0, and the line x = 0. Determine the surface area of the solid obtained by revolving R about the x axis.

5. (a) Compute the integral ∫_o^Txe^-xdx

where T > 0.

(b) Compute _limT→∞ ∫_o^Txe^-xdx