Find the area bounded by the curves y x3 and y 2x find

Question

1. Choose the most true statement and explain.

a. The integral is a "continuous sum".
b. The integral is the inverse of the derivative.
c. Both (A) and (B)
d. None of the above.

2. True or false and explain. In order to use integration by parts, we need to have a function that "diminishes"; i.e. a function that eventually has a derivative that is a constant or a derivative that can combine with the other function to be integrated normally.

Which of the following statements are true? Explain.

a. ∞ is a real number.
b. 1/0 = ∞.
c. 1/0 = ±∞
d. ∞ is not a real number.

4. Problems

a. Integrate ∫3 sin² (x) cos (x) dx

b. Use the disk method to find the volume of the solid formed by revolving the region bounded by y = 0.5x^2, x = 0, and y = 6 about the y-axis.

c. There is a mass of 14 at (-10, 0) and a mass of 26 at (6, 0). Find where a mass of 12 should be placed on the x-axis so that (5, 0) is the center of mass.

d. Find the area bounded by the curves y = 2x² - 5x - 8 and y = x + 12.

e. Find the area bounded by the curves y = x³ and y = 2x.

f. Find the distance between the points (4,-5,8) and (3,7,-2).

g. Let z = e^4x cos(xy). Find ∂z/∂x.

h. Let z = e^4x cos(xy). Find ∂z/∂y.

i. Let z = 2x² + 5y² - 8xy + 6x. Find the critical points of z.