A tank is of the shape of an inversed cone as shown in the

Question

1. Find the area of the region between the graphs of y = x² - 4x and y = 3 - 2x in the interval [0, 4].

2. Let R be the region above the graph of y = (x - 1)² and under the graph of y = x + 1.

(a) Find the volume of the solid obtained by revolving this region about the line y = 4.

(b) Find the volume of the solid by revolving this region about the y-axis.

3. A tank is of the shape of an inversed cone as shown in the following figure. It is filled with water of density ρ =1000 kg/m³. Find the work needed to pump the water in the tank to a point 2 meters above the top of the tank. (Use g = 9.81 m/sec²)

4. Use the definition to determine if each of the following improper integral is convergent or divergent. If it is convergent, find its value.

(a) ₁∫²1/√(x-1) dx.

(b) ₀∫^∞1/(x+2)(x+3).

5. Use comparison test to determine whether each of the following improper integral is convergent:

(a) ₁∫^∞x+1/√(x³+x)dx

(b) ₀∫¹x+1/√(x³+x)dx